The second most common Geometry type on the SAT (4.7% of the bank): angle chasing and similar triangles. Covers the angle sums that fill in a figure, vertical and corresponding angles across parallel lines, isosceles triangles, the exterior angle rule, matching corresponding sides of similar triangles by the angles they face, and the discipline of using only what is marked when a figure is not drawn to scale — with six worked examples, four traps and three checks.
Subject: SAT Prep · 61 slides · applied lesson
Open the interactive version of this deck
Title
SAT Math · Type 10 of 19
4.7% of the question bank — 78 of 1675 questions
Objectives
Two topics share this label. One is angle chasing: a figure with parallel lines or triangles, a couple of known angles, and a chain of deductions to the one you want. The other is similar triangles, where equal angles force the sides into a fixed ratio. Both reward marking the figure up rather than reasoning in your head.
One instruction covers most of the type: write on the picture. Angle chasing done in your head is where the deductions get lost.
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — 78 tagged questions of this type in the site's bank
Section
Section 1
Concept
Two topics share this label. One is angle chasing: a figure with parallel lines or triangles, a couple of known angles, and a chain of deductions to the one you want. The other is similar triangles, where equal angles force the sides into a fixed ratio. Both reward marking the figure up rather than reasoning in your head.
You will see it phrased in these ways:
The tell for the similar-triangle half is two triangles sharing an angle, which most often looks like one triangle nested inside another with a line parallel to one side.
Picture it
The same three-panel card as the survey deck, so the shorthand carries over: what identifies it, what you write first, and what is built to catch you.
Figure (svg): Lines, angles, and triangles: the tell, the move, and the trap
The red panel cuts both ways, and the second direction is worth money: when a figure is NOT labelled as out of scale, estimating from it is a legitimate way to eliminate choices.
Prediction
Knowing what a picture licenses is most of this type.
Predict first
A figure is marked not drawn to scale. Which of these may you still use?
Correct: A right-angle square drawn at a vertex
Why: A marked right angle is a stated fact, not an appearance, so it survives the not-to-scale warning. The other three are all appearances: looking like 90 degrees, looking equal, and looking like a midpoint are exactly what the warning tells you to disregard. The rule is simple — use what is marked or stated, and use nothing that is merely drawn.
Concept
Three shapes of question, and they need different tools.
| variant | what you use | how it usually looks |
|---|---|---|
| Angle chase | angle sums and parallel-line rules | a transversal, or several triangles sharing vertices |
| Similar triangles | proportional corresponding sides | one triangle nested in another, or a shared angle |
| Triangle properties | isosceles base angles, exterior angle rule | tick marks on equal sides, or an angle outside the triangle |
The angle chase is the most common, and it is the one that rewards marking up the figure most heavily, because each deduction feeds the next.
Definition probe
Naming the rule is the deduction.
Sort into buckets
Which rule gives you the missing angle?
Discrimination
Six things you might read off a not-to-scale figure.
Sort into buckets
May you use it?
Warm-up
Try it, marking the figure as you go.
Discussion prompt
Two parallel lines are cut by a transversal. One angle at the upper intersection is 118 degrees. What is the co-interior angle at the lower intersection, on the same side of the transversal?
Hint: Co-interior angles sit between the parallels on the same side.
Answer:
62 degrees. Co-interior angles, sometimes called same-side interior angles, always sum to 180 when the lines are parallel.
So the answer is 180 minus 118, which is 62.
The alternative route: the corresponding angle at the lower intersection is also 118, and the angle next to it on the straight line is 180 minus 118, which is 62. Two rules instead of one, same answer.
Both routes are fine, and having two is useful: if you cannot remember whether co-interior angles are equal or supplementary, the corresponding-then-straight-line route reconstructs it.
Pattern
Three steps, and the first involves writing rather than thinking.
Mark every given fact onto the figure: angles, tick marks, right angles, parallel arrows.
Why: The figure is your working memory. Deductions made in your head get lost two steps later, and a marked figure lets you see the next move rather than recall it.
Fill in every angle you can deduce in one step, even ones you do not obviously need.
Why: Angle chases proceed by chains, and the angle you did not need is frequently the bridge to the one you did.
For similar triangles, write the correspondence explicitly before writing any ratio.
Why: Which side matches which is decided by the angles they sit opposite, not by their position on the page.
Step 2 sounds wasteful and is not. Filling in the easy angles costs seconds and turns a search into a look.
Section
Section 2
Concept
A triangle totals 180, a straight line totals 180, and a full turn around a point totals 360.
| configuration | total | typical use |
|---|---|---|
| Interior angles of a triangle | 180 | find the third angle |
| Angles on a straight line | 180 | step from inside a figure to outside it |
| Angles around a point | 360 | several lines meeting at one vertex |
| Interior angles of a quadrilateral | 360 | two triangles joined |
Because they are unconditional, these are always the safest first deductions. Use them to fill the figure before reaching for anything that needs parallel lines.
Concept
When two lines cross, the angles opposite each other are equal, whether or not anything is parallel.
Whenever you see two lines cross, immediately fill in all four angles from the one you know. It is one of the cheapest deductions available.
Prediction
Two values in the whole figure.
Predict first
Parallel lines are cut by a transversal, and one angle is 73 degrees. Which value is NOT anywhere in the figure?
Correct: 90
Why: Every angle in such a figure is either equal to 73 or supplementary to it, giving 107. Since 73 is not 90, no right angle appears anywhere. This is the practical shortcut for parallel-line figures: mark one angle and there are only two distinct values in the entire picture, which turns most angle chases into a single subtraction.
Concept
Across a transversal, corresponding and alternate angles are equal, and co-interior angles sum to 180.
| pair | where they sit | relationship |
|---|---|---|
| Corresponding | same position at each parallel line | equal |
| Alternate interior | between the parallels, opposite sides | equal |
| Alternate exterior | outside the parallels, opposite sides | equal |
| Co-interior | between the parallels, same side | sum to 180 |
That last observation is the practical shortcut: mark one angle, and every other angle in the figure is either that value or 180 minus it.
Prediction
One angle determines the rest.
Predict first
An isosceles triangle has an apex angle of 36 degrees. What is each base angle?
Correct: 72
Why: The three angles total 180, so the two base angles share 180 minus 36, which is 144. They are equal, so each is 72. Answering 144 gives their combined total without halving it, which is the standard incompleteness on this rule. Answering 54 halves 108 instead, having subtracted from 180 incorrectly.
Concept
Two equal sides force the two angles opposite them to be equal, and the converse holds too.
The equal angles sit opposite the equal sides, not between them. Getting that correspondence right is the whole of the rule.
Concept
Extend one side of a triangle, and the angle formed outside equals the sum of the two interior angles not adjacent to it.
Remote means the two angles it does not touch. Adding the adjacent one instead is the standard misuse of this rule.
Prediction
Match by angle, not by position.
Predict first
Two similar triangles have every length of the larger 3 times the smaller. The smaller has a side of 5. What is the corresponding side of the larger?
Correct: 15
Why: Similar triangles scale every length by the same factor, so the corresponding side is 3 times 5, which is 15. The answer 5 over 3 divides instead of multiplying, which would be right if you were going from the larger to the smaller. The 45 applies the factor twice, which is the area scaling rather than the length scaling.
Concept
Equal angles make two triangles the same shape, so every pair of corresponding sides has the same ratio.
The commonest similar-triangle error is not the algebra, it is pairing the wrong sides. Write the correspondence out in words before writing any fraction.
Concept
When a figure says not drawn to scale, its appearance carries no information at all.
So the caption is worth reading before the figure. It tells you whether the picture is evidence or merely a diagram.
Prediction
Add the two it does not touch.
Predict first
A triangle has interior angles of 50 and 60 degrees. What is the exterior angle at the third vertex?
Correct: 110
Why: The exterior angle equals the sum of the two remote interior angles, which are 50 and 60, giving 110. Checking the other way: the third interior angle is 180 minus 110, which is 70, and the exterior angle on the straight line is 180 minus 70, which is 110. The answer 70 is the third interior angle, which is a genuine quantity and the wrong one.
Two truths and a lie
Three of these geometry statements are correct. The one left standing is false.
Eliminate the wrong options
Which statement is FALSE?
Survives elimination: c
Why: Similar triangles have PROPORTIONAL corresponding sides, not equal ones. Equal corresponding sides would make them congruent, which is the special case where the scale factor happens to be 1. The distinction matters because similar triangles are common on the SAT and congruent ones are rare, so assuming equality where only proportion holds is a reliable way to lose the question.
Check
Mark the figure as you go.
Check your understanding
Two parallel lines are cut by a transversal. One of the eight angles measures 47 degrees. Which of these could NOT be the measure of another angle in the figure?
Answer: A
Why: In a parallel-line figure every angle is either equal to the given one or supplementary to it, so only 47 and 133 appear, four times each. Since 94 is neither, it cannot occur anywhere in the figure.
The two-value fact turns this from a chase into a single subtraction. Any parallel-line figure contains exactly two distinct angle measures.
Section
Section 3
Worked example
Two parallel lines are cut by a transversal. One angle at the first intersection measures 118 degrees. Find the alternate interior angle and the co-interior angle at the second intersection.
Figure (svg): Two parallel lines cut by a transversal with angles marked at both intersections
Mark the given 118 on the figure, then mark its vertical partner, also 118.
Why: Vertical angles are equal and need no parallel lines, so this is the safest first deduction.
The alternate interior angle at the second intersection is also 118.
Why: Alternate interior angles across parallel lines are equal.
The co-interior angle is 180 minus 118, which is 62.
Why: Co-interior angles between parallels on the same side are supplementary.
The two-value observation is the fastest check available on this variant. If your figure ends up with three distinct angle values, something is wrong.
Verify: check that every angle in the figure is either 118 or 62, and that they sum to 180.
Why: Only two distinct values can appear in a parallel-line figure, which confirms the deductions are consistent.
Answer: alternate interior 118 degrees; co-interior 62 degrees
Worked example
A triangle has two sides marked with tick marks and an apex angle of 44 degrees. Find the base angles.
Figure (svg): A triangle with two sides marked equal and the apex angle labelled
The tick marks say two sides are equal, so the angles opposite them are equal.
Why: This is the isosceles property, and the tick marks are a stated fact rather than an appearance.
The three angles total 180, so the two base angles share 180 minus 44, which is 136.
Why: Subtracting the known apex leaves the total of the other two.
They are equal, so each is 136 divided by 2, which is 68.
Why: Halving the remaining total gives each base angle.
The equal angles sit OPPOSITE the equal sides. Placing them between the equal sides is the standard misapplication of the rule.
Verify: check the sum: 44 plus 68 plus 68 is 180.
Why: The interior angles total 180, confirming the split.
Answer: each base angle is 68 degrees
Worked example
A line parallel to the base of a triangle cuts the other two sides, creating a smaller triangle at the top. The small triangle has base 4 and the large has base 10. The small triangle's left side is 6. Find the large triangle's left side.
Figure (svg): A small triangle nested inside a larger one, with corresponding sides labelled
The parallel line makes corresponding angles equal, and the apex angle is shared, so the triangles are similar.
Why: Two pairs of equal angles is sufficient for similarity.
Write the correspondence: the bases 4 and 10 match, and the left sides 6 and x match.
Why: Corresponding sides face corresponding angles, and the parallel cut makes the matching obvious here.
Set up small over large consistently: 4 over 10 equals 6 over x, so 4x equals 60 and x equals 15.
Why: Both sides of the equation use small on top, which is what keeps the proportion valid.
Computing the scale factor first is often the cleanest route: find it once, then multiply or divide every length by it.
Verify: check the scale factor: 10 over 4 is 2.5, and 6 times 2.5 is 15.
Why: Every length scales by the same factor, confirming the answer independently of the cross-multiplication.
Answer: the large triangle's left side is 15
Step zero
Before any deduction.
Discussion prompt
A question shows a figure with parallel lines, a triangle, and one labelled angle, then asks for a different angle. What is the first thing you do?
Hint: It involves your pencil rather than your reasoning.
Answer:
Mark the given angle onto the figure, then fill in every angle you can deduce in one step.
Its vertical partner, its supplement on the straight line, and its corresponding angle across the parallels all follow immediately.
Do this even for angles you do not think you need. Angle chases work in chains, and the bridging angle is rarely the one you would have predicted.
In a parallel-line figure there are only two distinct values, so filling in the whole picture takes a few seconds and usually reveals the answer without further work.
The reason it matters: deductions held in your head are lost after two steps. A marked figure turns the search into a look.
Worked example
In a triangle, two interior angles are 43 and 62 degrees. One side is extended at the third vertex. What is the exterior angle there?
Figure (svg): A table deriving the exterior angle two different ways
Apply the exterior angle rule directly: it equals the two remote interior angles, 43 plus 62.
Why: The remote angles are the two it does not touch.
That gives 105 degrees.
Why: Adding the two given angles is the whole computation.
Cross-check via the long route: the third interior angle is 180 minus 105, which is 75, and 180 minus 75 is 105.
Why: Going via the straight line reproduces the same answer.
If you cannot recall the exterior angle rule, derive it via the third interior angle and the straight line. It costs one extra step and cannot be misremembered.
Verify: check that the exterior angle exceeds each remote interior angle.
Why: It is their sum, so it must be larger than either — a quick sanity check on the direction.
Answer: 105 degrees
Worked example
A figure marked not drawn to scale shows a triangle with sides labelled 5, 12 and 13, and one angle appears to be obtuse. Is the triangle right-angled?
Figure (svg): A right triangle with sides five, twelve and thirteen
Ignore the appearance entirely, since the figure is not drawn to scale.
Why: An angle that looks obtuse carries no information under that warning.
Test the side lengths against the Pythagorean relationship: 5 squared plus 12 squared is 25 plus 144, which is 169.
Why: The Pythagorean theorem is on the reference sheet and applies to the stated lengths.
And 13 squared is also 169, so the relationship holds and the triangle is right-angled.
Why: Equality of the two sides of the relationship is exactly the condition for a right angle.
This is the not-to-scale rule working in your favour: the stated lengths overrode a misleading picture and settled the question completely.
Verify: check that 5, 12, 13 is a standard triple worth recognising on sight.
Why: Alongside 3, 4, 5 it is the pair the SAT reuses most, so recognising it saves the computation.
Answer: yes, it is right-angled, despite how it is drawn
Faded example
From memory. Four facts that fill in most figures.
Fill in the blanks
The interior angles of a triangle sum to 180. Angles on a straight line sum to 180. Vertical angles are equal, whether or not any lines are parallel. And across parallel lines, co-interior angles sum to 180.
Why: Three of these four are 180, which is worth noticing: almost every angle deduction on this type is either a transfer of a value or a subtraction from 180. That is why a parallel-line figure only ever contains two distinct angle values.
Worked example
Two parallel lines are cut by a transversal, forming a triangle with a third line. One angle at the upper parallel is 55 degrees, and the triangle has another angle of 65 degrees. Find the third angle of the triangle.
Figure (svg): Parallel lines cut by two transversals forming a triangle
Transfer the 55 degrees across the parallel lines using alternate angles, so the triangle has an angle of 55.
Why: Alternate angles across parallel lines are equal, which moves a known value to where it is useful.
The triangle now has two known angles, 55 and 65.
Why: One was transferred and one was given directly.
The third is 180 minus 55 minus 65, which is 60 degrees.
Why: Interior angles of a triangle total 180.
The transfer step is what makes this a chase rather than a single rule. Marking the transferred angle on the figure is what makes the triangle sum visible.
Verify: check the sum: 55 plus 65 plus 60 is 180.
Why: The three angles total 180, confirming the chain of deductions.
Answer: 60 degrees
Fill the middle
What similarity gives you, and what it does not.
Fill in the blanks
Two triangles are similar when two pairs of angles are equal. Their corresponding sides are then proportional, not equal. Corresponding sides are matched by the angles they sit opposite. If every length of the larger is 3 times the smaller, its area is 9 times the smaller.
Why: The last blank connects this type to type 8. Similar triangles are a scaling situation, so lengths scale by k and areas by k squared. A question about the ratio of areas of two similar triangles is asking about k squared, and answering with k is the standard error.
Estimation
When a figure is NOT marked out of scale, it is drawn accurately.
Predict first
A figure with no scale warning shows an angle that looks a little less than a right angle. Which choice is most likely correct?
Correct: 82 degrees
Why: With no not-to-scale warning the figure is drawn accurately, so an angle that looks a little under 90 is a little under 90. That eliminates the two obtuse choices immediately and makes 38 implausible, since it would look obviously acute. This is the underused half of the scale rule: an accurate figure is evidence, and estimating from it is a legitimate way to eliminate.
Check
Find the scale factor first.
Check your understanding
Two similar triangles have corresponding sides of 6 and 15. If another side of the smaller triangle is 8, what is the corresponding side of the larger?
Answer: A
Why: The scale factor is 15 over 6, which is 2.5. Applying it to the side of 8 gives 8 times 2.5, which is 20. Checking with a proportion: 6 over 15 equals 8 over x, so 6x equals 120 and x equals 20.
Choice B is worth noticing: similarity scales by multiplication, never by addition, and the additive answer is offered on nearly every question of this shape.
Section
Section 4
Trap
The trap. A triangle is drawn with one angle that looks square, so you use the Pythagorean theorem and everything after that is clean.
Underneath, in small grey type, are the words not drawn to scale. There was no right angle, and the theorem never applied.
The same trap runs on lengths: a side that looks twice another is not, and a point that looks like a midpoint is not.
The fix. Read the caption before the figure, and treat the drawing as a diagram of relationships rather than a measurement.
Use only what is marked: a right-angle square, tick marks, arrows, or a stated number.
Trap
The trap. Two similar triangles are drawn in different orientations, and you match the sides by their position on the page — leftmost with leftmost, longest with longest.
Corresponding sides are decided by the ANGLES they sit opposite, and a rotated or reflected triangle puts them in different places.
The resulting proportion looks perfectly reasonable and gives a wrong length.
The fix. Write the correspondence out in words before writing any fraction.
Say: the side opposite the 40-degree angle in the small triangle matches the side opposite the 40-degree angle in the large one.
Error analysis
A student's proportion for two similar triangles. The setup is inconsistent.
Annotate
On: \( \frac{\text{small base}}{\text{large base}} = \frac{\text{large side}}{\text{small side}} \)
Whenever a proportion can be set up two ways, prefer the scale factor. It replaces a positional decision with a single multiplication.
Trap
The trap. A figure shows two lines cut by a transversal and you use corresponding angles, concluding two angles are equal.
But nothing marked the lines as parallel. Corresponding, alternate and co-interior all REQUIRE parallelism, and without it none of them hold.
Vertical angles and the angle sums still work, but those are the only tools available.
The fix. Look for the arrows. Parallel lines are marked with matching arrowheads, or stated in words.
If they are absent, restrict yourself to vertical angles and the sums.
Elimination
A triangle has interior angles of 80 and 55 degrees.
Eliminate the wrong options
What is the exterior angle at the third vertex? Three choices can be ruled out by reasoning.
Survives elimination: b
Why: The exterior angle equals the sum of the two remote interior angles, 80 plus 55, which is 135. Note how much was eliminated without arithmetic: an exterior angle must be less than 180 and greater than either remote interior angle, which removes three choices on structure alone. Building those bounds into your instinct makes this variant nearly automatic.
Trap
The trap. A triangle has angles 43 and 62 and you compute the exterior angle at the third vertex as 62 plus 75, or as 180 minus 43.
The exterior angle equals the two remote interior angles — the two it does not touch — which are 43 and 62, giving 105.
Including the adjacent angle double-counts and gives a number greater than 180, which is impossible for an exterior angle of a triangle.
The fix. Identify the vertex where the exterior angle sits, then add the OTHER two interior angles.
Or derive it: find the third interior angle and subtract from 180.
Counterexample
A rule students over-extend.
Discussion prompt
A student says: if two triangles have the same angles, they have the same size. Give a counterexample and name the distinction being missed.
Hint: Think about a photograph and its enlargement.
Answer:
Counterexample: a 3-4-5 triangle and a 6-8-10 triangle. Both have exactly the same three angles, and one is twice the size of the other.
The distinction being missed is between similar and congruent. Equal angles give the same SHAPE, which is similarity. Same size as well requires equal sides, which is congruence.
Why angles cannot fix size: scaling every length by the same factor leaves every angle unchanged. A photograph enlarged is the same shape and a different size.
On the SAT this matters because similar triangles are common and congruent ones are rare. Assuming equal sides where only proportion holds is a reliable way to lose the question.
And the useful corollary: since the angles are equal, you can always find the scale factor from any one pair of corresponding sides, then apply it to all the others.
Edge cases
The warning removes information. Its absence adds some.
Discussion prompt
What exactly may you use when a figure IS marked not to scale, and what changes when that caption is absent?
Hint: The second half is the part students never exploit.
Answer:
When it IS marked not to scale: use only stated facts — labelled measurements, right-angle squares, tick marks for equal lengths, arrows for parallel lines, and anything asserted in the text.
Disregard every appearance: angles that look right, sides that look equal, points that look collinear or midway.
When the caption is ABSENT, the figure is drawn accurately. That is a guarantee, and it means estimating from the picture is legitimate.
So you can measure an angle by eye, compare two lengths, and eliminate choices that are visibly the wrong size.
The practical upshot: on an accurate figure, estimation is a valid elimination tool and frequently gets you to one answer without any geometry at all.
The one caution: estimation eliminates, it does not confirm. If two choices are close in size, you still need the deduction.
Check
Read the caption before the figure.
Check your understanding
A figure marked not drawn to scale shows a triangle with two sides bearing tick marks and one angle labelled 50 degrees at the apex between them. What are the other two angles?
Answer: A
Why: The tick marks are a stated fact, so the triangle is isosceles and the two angles opposite the equal sides are equal. They share 180 minus 50, which is 130, so each is 65. The not-to-scale warning removes nothing here, because every fact used was marked rather than drawn.
Choice D is the over-correction: students who take the warning too far conclude that nothing can be determined. Marked facts always survive it.
Section
Section 5
Matching
Six situations, six rules. No arithmetic.
Match the pairs
Why: Only two of these six need the lines to be parallel. That is worth internalising, because the commonest structural error on this type is reaching for a parallel-line rule in a figure where nothing is marked parallel.
Sorting
The figure is marked not drawn to scale.
Sort into buckets
May you use this fact?
Four of the six are usable, which is the point worth taking away: not-to-scale removes far less than students fear, and the over-correction of deciding nothing can be determined is its own error.
Comparison
Fill the blanks from memory.
Comparison matrix
| similar | congruent | |
|---|---|---|
| Angles | equal | equal |
| Sides | proportional, in a fixed ratio | equal |
| Shape and size | same shape, any size | same shape and same size |
| How often on the SAT | common | rare |
Congruence is the special case of similarity where the scale factor is 1. Since the SAT overwhelmingly tests the general case, assume proportional rather than equal unless equality is marked.
Trade off
Fill in what each costs.
Comparison matrix
| method | steps | where it goes wrong |
|---|---|---|
| Set up a cross-multiplied proportion | write four sides, cross-multiply, solve | inverting one side of the equation |
| Find the scale factor, then multiply | divide one pair, then multiply the rest | multiplying when you should divide |
| Match sides by position on the page | no steps, and no reliability | a rotated triangle breaks it entirely |
| Match sides by the angle they face | mark the angles, then label the sides | nothing — it is the correct method |
The scale-factor route has one number to get right instead of four positions, which is why it is the safer of the two working methods. The bottom two rows are not alternatives — one is the correct way to identify correspondence and the other is the trap.
Real world
One minute on why similar triangles are genuinely useful.
Discussion prompt
Similar triangles are how people measure things they cannot reach. How does the shadow method work, and what makes it valid?
Answer:
Stand a metre stick next to a tree and measure both shadows. The sun's rays arrive at effectively the same angle for both, so the two triangles have equal angles and are similar.
That means stick height over stick shadow equals tree height over tree shadow, and the only unknown is the tree's height.
This is how the height of the Great Pyramid was first measured, and it is still how surveyors work with a theodolite.
The same principle underlies map scales, architectural models and camera optics. In every case, equal angles force a fixed ratio between lengths.
And it is why the SAT keeps testing correspondence rather than arithmetic: the whole method depends on matching the right side with the right side, and everything else is one division.
Ranking
A triangle has interior angles of 35 and 85 degrees. Order these four quantities.
Put in order
Why: The third interior angle is 180 minus 35 minus 85, which is 60. The exterior angle at each vertex is 180 minus that vertex's interior angle: at the 35-degree vertex it is 145, at the 85-degree vertex it is 95, and at the third vertex it is 120. Ordered smallest to largest that gives 60, then 95, then 120, then 145. Note the inverse relationship worth carrying: the smaller the interior angle, the larger its exterior angle.
Concept
This type is 4.7 per cent of the section and splits into two halves that need different practice.
| session | what you do | why |
|---|---|---|
| 1 | Fifteen angle chases, marking every deducible angle on the figure before answering. | Builds the marking habit, which is what makes chains of deductions reliable. |
| 2 | Ten parallel-line figures, checking every time that the lines are actually marked parallel. | Prevents the most common structural error in the type. |
| 3 | Twelve similar-triangle questions, writing the correspondence in words before any fraction. | Correspondence, not arithmetic, is where these questions are lost. |
| 4 | Ten figures with and without the not-to-scale caption, stating what may be used from each. | Trains both directions: what the warning removes, and what its absence permits. |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — filter the bank to this skill tag — 78 questions, roughly a third of each difficulty
Explain it to yourself
Close the deck.
Discussion prompt
Without looking, state what you may and may not use from a figure marked not drawn to scale, and what changes when that caption is absent.
Hint: Both directions carry information.
Answer:
When marked not to scale, use only stated facts: labelled measurements, right-angle squares, tick marks, parallel arrows, and anything asserted in the text.
Disregard appearances: angles that look right, sides that look equal, points that look collinear.
When the caption is absent, the figure is accurate, so estimating from it is legitimate and is a valid way to eliminate choices.
Estimation eliminates but does not confirm — if two choices are close in size, the deduction is still required.
If you gave both directions, you have the rule as the test actually uses it rather than only its cautionary half.
Explain it
Two minutes, out loud.
Discussion prompt
A friend matches similar-triangle sides by which looks longest. How do you show them the right method?
Answer:
Draw one triangle, then draw the same triangle rotated and scaled. Ask them which side matches which. Position no longer helps at all.
Then mark the angles. Show that the 40-degree angle in one triangle corresponds to the 40-degree angle in the other, wherever it has ended up on the page.
Give the rule as a sentence: a side corresponds to the side facing the same angle. Match by angle, never by position.
Then give the practical shortcut: label each side by the angle it faces before writing any proportion. The labelling does the matching for you.
And the safer route overall: find the scale factor from one known pair, then multiply. One number to get right instead of four positions.
Commit first
Commit before you check.
Predict first
Two similar triangles have areas in the ratio 9 to 25. What is the ratio of their corresponding sides?
Correct: 3 to 5
Why: Areas scale by the square of the length ratio, so to go from areas back to lengths you take square roots: the square root of 9 is 3 and of 25 is 5. Answering 9 to 25 confuses the area ratio with the length ratio, and 81 to 625 squares again rather than un-squaring. This is the type 8 scaling rule applied in reverse, which is the direction the SAT most often asks for.
Connect it up
Blank paper.
Draw it
Draw two parallel lines cut by a transversal and label all eight angles with only two values, showing that every angle is either the given one or its supplement. Beside it, draw a triangle and mark the three sums: 180 inside, 180 on a line, 360 around a point. Underneath, draw two similar triangles in different orientations and connect corresponding sides with arrows, labelling each side by the angle it faces. In a box at the side, write the two halves of the scale rule: not to scale means use only what is marked, and no caption means the figure is accurate so you may estimate. Finally list the four traps: trusting the picture, pairing sides by position, using parallel rules without parallel lines, and adding the adjacent angle to an exterior angle.
Exit ticket
One question before you close the deck.
Predict first
A figure shows two lines cut by a transversal, with no arrows and no statement about parallelism. Which rule may you use?
Correct: Vertical angles are equal
Why: Vertical angles are equal at any crossing, whether or not the lines are parallel — the rule follows from the straight-line sum alone. The other three all require parallelism, and nothing in this figure asserts it. Using them here would be reasoning from an assumption the question never made, which is the same error as trusting an unmarked right angle.
Recap
One type, one instruction: write on the picture, and use only what is stated.
| never do this | do this instead |
|---|---|
| Assume a right angle because it looks square | Use it only if a square is marked or it is stated |
| Match similar sides by position on the page | Match by the angle each side faces |
| Use corresponding angles without parallel markings | Fall back on vertical angles and the sums |
| Add the adjacent angle to get an exterior angle | Add the two remote interior angles |
| Conclude nothing can be determined from a not-to-scale figure | Marked facts always survive the warning |
| Give the length ratio when asked about areas | Square it — or take the root going the other way |
Naruhodo Tutoring — SAT Math question bank export (sat-question-index.json) — and the site's SAT pages to drill this type in isolation, then mixed
Want this taught 1-on-1? Alexander tutors SAT Prep — $55/session, free consultation.