The continuation of the Reveal Geometry Chapter 1 deck, picking up where the segment material stopped. It is organised around one parallel made explicit throughout: angles behave exactly like segments with degrees in place of length, so the angle addition postulate is the segment addition postulate and a bisector is a midpoint. It covers naming angles and when the vertex-only name is illegal, classification by measure, congruence marks as the only reliable information in a diagram, the angle addition postulate and bisectors with algebraic measures, adjacent angles, linear pairs, vertical angles, complementary and supplementary pairs, perpendicular lines, and finishes with polygons, perimeter, circumference and area including on the coordinate plane using the distance formula from the first deck.
Subject: Honors Geometry · 60 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
Honors Geometry - Chapter 1, second half
Everything you already know about segments, with degrees instead of length
Objectives
Last session finished with partitioning a directed segment. This deck picks up at angle measure and finishes the chapter.
One idea makes this half of the chapter much shorter than it looks, and it is worth stating before anything else.
McGraw Hill, Reveal Geometry — Module 1, Tools of Geometry Module 1 — The lessons this deck follows.
Concept
Every major idea in this deck is an idea from the first deck with degrees in place of length. Recognising the parallel means most of the work is already done.
| segments, last deck | angles, this deck |
|---|---|
| segment measure, written with a length | angle measure, written in degrees |
| segment addition postulate | angle addition postulate |
| congruent segments have equal lengths | congruent angles have equal measures |
| a midpoint splits a segment into two congruent parts | a bisector splits an angle into two congruent parts |
If a problem about angles looks unfamiliar, rewrite it as the matching problem about segments. The algebra is identical and you have already done it.
Picture it
The same postulate, twice, in the two settings.
Figure (svg): Side by side: a segment with three labelled points illustrating the segment addition postulate, and an angle with an interior ray illustrating the angle addition postulate.
Keep this picture in mind for the whole deck. Whenever a new angle result appears, ask what the segment version was.
Section
What an angle is, and how to name it without ambiguity
Concept
The two rays are the sides of the angle, and the point they share is the vertex. That is the whole definition.
Vertex — The common endpoint of the two rays forming an angle.
Sides — The two rays themselves. They are rays and not segments, so they have no length and extend forever.
Because the sides are rays, the size of an angle has nothing to do with how long the rays are drawn. Two angles can look very different on the page and have exactly the same measure.
Prediction
Two angles are drawn, both opening by the same amount. The rays of the first are drawn twice as long as the rays of the second.
Predict first
How do their measures compare?
Correct: They are equal
Why: An angle measures the amount of turn between two rays, and a ray already extends forever. How much of it you draw is a matter of ink, not of geometry. This is the single most common misconception about angle measure, and it comes from thinking of the sides as segments rather than rays.
Concept
Every angle can be named in more than one way, and one of the ways is only sometimes legal.
\[ \angle ABC \quad \text{or} \quad \angle CBA \quad \text{-- same angle, vertex } B \text{ in the middle} \]
Reversing the two outer letters names the same angle. Moving the vertex letter out of the middle names something else entirely, or nothing at all.
Picture it
A ray drawn inside an angle creates three angles at the same vertex, not one.
Figure (svg): An angle ABC with an additional ray BD drawn inside it, creating three separate angles that all share vertex B.
Whenever a diagram has a ray drawn inside an angle, the vertex-only name is off the table for every angle at that vertex.
Trap
The diagram has vertex B, so the angle can be called angle B. That is shorter and everybody will know what is meant.
\[ \angle B \]
Only when exactly one angle has that vertex. As soon as a second ray is drawn from B, there are three different angles at B and the name picks out none of them.
\[ \angle ABD, \quad \angle DBC, \quad \angle ABC \quad \text{-- three different measures} \]
Graders treat this as wrong rather than as informal, because the answer genuinely does not say which angle it refers to. On a proof it can make an otherwise correct argument unmarkable.
The safe habit is to use three letters always. It costs two characters and it is never ambiguous, and it also forces you to identify the vertex, which is the part students most often get wrong.
Concept
An angle is measured by how far one side is rotated from the other, in degrees, and the measure is always taken as the amount between 0 and 180 for the angles in this chapter.
\[ m\angle ABC = 62 \]
The letter m in front means the measure of, so the expression above is a number. The angle itself is a geometric figure and the measure of it is a number, and keeping those apart matters when you write equations.
| classification | measure |
|---|---|
| acute | between 0 and 90 |
| right | exactly 90 |
| obtuse | between 90 and 180 |
| straight | exactly 180 |
Notation
This distinction is the angle version of one you already met with segments.
Annotate
On: \( \angle ABC \cong \angle DEF \qquad \text{versus} \qquad m\angle ABC = m\angle DEF \)
Rule of thumb: congruent goes with figures, equals goes with numbers. If the letter m is present you are talking about a number, so use an equals sign.
Definition probe
Sort by the classification, using the boundaries exactly.
Sort into buckets
Sort each measure by its classification.
Concept
A diagram tells you only what its marks say. Matching arcs on two angles mean those angles are congruent, and a small square at a vertex means that angle is exactly 90 degrees.
Anything not marked is not known, however it looks. An angle that appears to be a right angle is not one unless the square is drawn or the problem says so.
This is the same rule as the tick marks on congruent segments from the first deck, and it is enforced just as strictly.
Two truths and a lie
Four claims about an unmarked diagram. One is safe.
Eliminate the wrong options
Rule out the three that read more than the diagram says, and keep the safe one.
Survives elimination: r3
Why: Betweenness is one of the few things a diagram genuinely does communicate: which ray is drawn inside the angle is part of the configuration rather than a measurement. Sizes, congruences and right angles all have to be marked or given, because the drawing is not to scale.
Warm-up
Recall this from the first deck before meeting its angle twin, because the two are the same problem.
\[ \text{B is between A and C: } \; AB = 2x + 3, \; BC = x - 1, \; AC = 17 \]
Discussion prompt
Set up and solve for x, then say in words which postulate you used.
Hint: The two parts total the whole.
Answer:
\[ (2x + 3) + (x - 1) = 17 \;\Longrightarrow\; 3x + 2 = 17 \;\Longrightarrow\; x = 5 \]
The segment addition postulate. Checking: the first part is 13 and the second is 4, totalling 17. Everything in the next part of this deck is this exact problem with degrees instead of lengths.
Explain it to yourself
Naming is not usually where the difficulty of a problem lives, and it is often where the marks go.
Discussion prompt
Explain why writing three letters for every angle protects you even in diagrams where the vertex-only name would be legal.
Hint: Think about what happens later in a problem when a new ray gets drawn.
Answer:
Diagrams grow. A problem that begins with a single angle at B often adds a bisector or an auxiliary ray partway through, and every vertex-only name written earlier becomes ambiguous at that moment.
Three letters also force you to identify the vertex explicitly, which is the part students most often get wrong. Writing angle ABC makes you decide that B is the vertex before you write anything else.
It costs two characters and removes an entire category of error, which is about as good a trade as geometry notation offers.
Pattern
A short checklist for the start of any angle problem, before any algebra begins.
Step three is the one that separates a careful answer from a plausible one. Most wrong geometry answers are correct reasoning applied to information the diagram never provided.
Section
The postulate you already know, in a new setting
Concept
If a ray is drawn from the vertex into the interior of an angle, it splits that angle into two, and the two parts total the whole.
\[ m\angle ABD + m\angle DBC = m\angle ABC \]
The word interior is doing real work. The postulate only applies when the ray is genuinely inside the angle, which is exactly the betweenness condition from the segment version.
The parallel is worth repeating: this is the segment addition postulate with degrees instead of lengths, and the algebra it produces is identical.
Worked example
Ray BD is drawn inside angle ABC. The parts and the whole are given as expressions and a number.
\[ m\angle ABD = 3x + 5, \quad m\angle DBC = 2x - 3, \quad m\angle ABC = 62 \]
Write the postulate before substituting anything.
Why: Writing the general statement first keeps the parts and the whole in the right places, which is where most errors in these problems occur.
\[ m\angle ABD + m\angle DBC = m\angle ABC \]
Substitute the expressions and collect like terms.
Why: The two parts are added on the left and the whole sits on the right, exactly as the postulate says.
\[ (3x + 5) + (2x - 3) = 62 \;\Longrightarrow\; 5x + 2 = 62 \]
Solve for the unknown.
Why: Subtracting 2 and dividing by 5 is ordinary equation solving, which is why the geometry here is really just careful setup.
\[ 5x = 60 \;\Longrightarrow\; x = 12 \]
Figure (svg): An angle ABC with a ray BD drawn inside it, splitting the angle into two parts labelled with algebraic expressions.
Verify: substitute the value back into both parts.
Why: The first part is 3 times 12 plus 5, which is 41. The second is 2 times 12 minus 3, which is 21. Adding gives 62, matching the whole angle exactly, so the value of x is right and so was the setup.
Concept
A ray that splits an angle into two congruent angles is called the bisector of that angle.
Angle bisector — A ray from the vertex, into the interior, that divides the angle into two angles of equal measure.
\[ \text{if } \overrightarrow{BD} \text{ bisects } \angle ABC, \text{ then } m\angle ABD = m\angle DBC \]
Because the two parts are equal, a bisector problem gives you an equation immediately: set the two parts equal to each other. That is a different equation from the addition postulate, and choosing the right one is the whole skill.
Worked example
Ray BD bisects angle ABC, and the two parts are given as expressions.
\[ m\angle ABD = 4x - 6, \qquad m\angle DBC = 2x + 18 \]
Use the definition of a bisector to write the equation.
Why: Bisected means the two parts have equal measures, so the two expressions are set equal to each other rather than added.
\[ 4x - 6 = 2x + 18 \]
Solve the equation.
Why: Collecting the variable terms on one side and the numbers on the other is ordinary algebra.
\[ 2x = 24 \;\Longrightarrow\; x = 12 \]
Answer the question that was actually asked.
Why: Problems usually want an angle measure rather than the value of x, and stopping at x is the most common way to lose marks on a correct solution.
\[ m\angle ABD = 4(12) - 6 = 42, \qquad m\angle ABC = 2 \times 42 = 84 \]
Figure (svg): An angle ABC with ray BD bisecting it, both halves marked as congruent with matching arcs.
Verify: check both halves are equal and that they total the whole.
Why: The first half is 4 times 12 minus 6, which is 42. The second is 2 times 12 plus 18, which is also 42. They are equal, as a bisector requires, and together they give 84 for the whole angle.
Discrimination
The two setups look similar and produce different equations. Choosing wrongly makes the rest of the work irrelevant.
Sort into buckets
Sort each situation by which equation it produces.
Fill the middle
Ray QS bisects angle PQR. The two halves are 5x minus 4 and 3x plus 10.
Fill in the blanks
5x - 4 = 3x + 10 \;\Longrightarrow\; x = 7
Why: A bisector makes the two halves equal, so the two expressions are set equal to each other. Collecting terms gives 2x equals 14, so x is 7. Checking: 5 times 7 minus 4 is 31, and 3 times 7 plus 10 is also 31, so both halves measure 31 and the whole angle is 62.
Error analysis
The bisector problem again. The algebra is flawless and the submission still loses marks.
Annotate
On: \( 4x - 6 = 2x + 18 \;\Longrightarrow\; x = 12 \;\Longrightarrow\; \text{answer: } 12 \)
Re-read the question after solving. Geometry problems almost always want a measure, and x is only ever an intermediate step on the way to one.
Pattern
Nearly every algebraic angle problem in this chapter is one of four equations. Identifying which one is the entire difficulty.
| the problem tells you | write this |
|---|---|
| a ray is drawn inside the angle | part plus part equals whole |
| the ray bisects, or the parts carry matching arcs | part equals part |
| the angles form a linear pair | the two measures total 180 |
| the angles are vertical | the two measures are equal |
The final check is not optional politeness. It catches a wrong row choice immediately, because the numbers will fail the relationship you claimed.
Anomaly
A ray BD is drawn inside angle ABC. A student writes the following and solves it.
\[ (3x + 5) + (2x - 3) = (4x + 1) \]
Predict first
Something is structurally wrong before any solving happens. What?
Correct: The right side is one of the parts, not the whole angle
Why: The angle addition postulate puts the two parts on one side and the whole on the other. If the expression on the right is itself one of the pieces of the diagram rather than the full angle ABC, the equation is stating that two parts total a third part, which the postulate never says. Identifying which measure is the whole, before substituting, is what prevents this.
Explain it
A classmate is told ray BD bisects angle ABC, with the halves given as 4x minus 6 and 2x plus 18, and the whole angle given as 84.
\[ (4x - 6) + (2x + 18) = 84 \]
Discussion prompt
Their equation is not wrong, but they say it contradicts yours. Explain what is going on.
Hint: Can two different correct equations come from one diagram?
Answer:
Both equations are valid. Yours used the bisector to set the halves equal; theirs used the angle addition postulate to make the halves total the whole. A bisected angle satisfies both relationships at once.
\[ 6x + 12 = 84 \;\Longrightarrow\; x = 12 \]
Their equation gives x equals 12, exactly as yours did. When a diagram supplies two facts, either can be used, and getting the same answer from both is the strongest check available.
Section
Pairs of angles with names, and what each name promises
Concept
Two angles are adjacent when they share a vertex and a side, and their interiors do not overlap.
All three conditions matter. Two angles sharing only a vertex are not adjacent, and neither are two angles where one sits inside the other.
Adjacency on its own promises nothing about the measures. It is a description of position, not a relationship between numbers, which is why it is never a reason in a proof by itself.
Concept
Two named configurations do most of the work in this chapter, and only one of them is about position alone.
Linear pair — Two adjacent angles whose non-shared sides form a straight line. Their measures always total 180.
Vertical angles — The two opposite angles formed where two lines cross. They are always congruent.
\[ \text{linear pair: } m\angle 1 + m\angle 2 = 180 \qquad \text{vertical: } \angle 1 \cong \angle 3 \]
Both of these are facts you may use without proving them, and both are extremely common as the first step of a longer problem.
Picture it
Every relationship in this part is visible in this one picture.
Figure (svg): Two lines crossing at a point, forming four numbered angles, with opposite pairs identified as vertical angles and neighbouring pairs as linear pairs.
Knowing any one of these four measures gives you all four. That is why this configuration appears at the start of so many problems.
Prediction
Two lines cross. One of the four angles measures 35 degrees.
Predict first
What are the other three measures?
Correct: 35, 145 and 145
Why: The angle opposite the 35 is vertical to it and therefore also 35. Each of the two neighbouring angles forms a linear pair with the 35, so each measures 180 minus 35, which is 145. The four angles are 35, 145, 35 and 145 going round, and they total 360 as angles around a point must.
Concept
Two more names, and these are about the numbers rather than the picture. The two angles do not have to be next to each other or even in the same diagram.
| name | measures total | memory hook |
|---|---|---|
| complementary | 90 | C comes before S, and 90 comes before 180 |
| supplementary | 180 | S for straight, and a straight angle is 180 |
Since a linear pair always totals 180, every linear pair is supplementary. The reverse does not hold: two angles can total 180 while sitting in completely different places.
Trap
These two angles are supplementary, so they form a linear pair and their outer sides make a straight line.
\[ m\angle P + m\angle Q = 180 \;\Longrightarrow\; \text{linear pair} \]
Supplementary is a statement about two numbers totalling 180. Linear pair is a statement about a configuration, and it says the angles are adjacent with their outer sides forming a line.
An angle of 130 in one diagram and an angle of 50 in another are supplementary and are not adjacent, do not share a vertex, and form no line at all.
The implication runs one way only: a linear pair is always supplementary, because the outer sides form a straight angle. Supplementary angles are only sometimes a linear pair.
The same one-way trap appeared in this chapter already, when a diagram looking like a right angle did not make it one. A property does not imply the configuration that usually produces it.
Worked example
Two angles form a linear pair and are given as expressions.
\[ (3x + 10) \quad \text{and} \quad (5x - 30) \]
Write the relationship the configuration guarantees.
Why: A linear pair always totals 180 degrees, so that is the equation, and it comes from the configuration rather than from anything given in numbers.
\[ (3x + 10) + (5x - 30) = 180 \]
Collect and solve.
Why: Combining like terms gives a simple linear equation.
\[ 8x - 20 = 180 \;\Longrightarrow\; 8x = 200 \;\Longrightarrow\; x = 25 \]
Find the two measures, since those are what the question wants.
Why: Substituting back into each expression gives the two angles.
\[ 3(25) + 10 = 85, \qquad 5(25) - 30 = 95 \]
Figure (svg): A ray drawn from a point on a straight line, forming two angles that together span the line and total 180 degrees.
Verify: check the two measures total 180.
Why: Adding 85 and 95 gives exactly 180, which is what a linear pair requires. If the total had come out as anything else, either the equation or the arithmetic would be wrong.
Faded example
Two angles are complementary. One is 2x and the other is 3x plus 15.
Fill in the blanks
2x + (3x + 15) = 90 \;\Longrightarrow\; x = 15
Why: Complementary angles total 90, not 180, and that is the only difference from the linear pair problem. Collecting gives 5x plus 15 equals 90, so 5x is 75 and x is 15. The two angles are 30 and 60, which total 90 as required.
Matching
Each description matches exactly one name.
Match the pairs
Why: Two of these names describe a numerical relationship and two describe a configuration. Complementary is purely about totalling 90. Adjacent is purely about position and promises nothing about measures. Linear pair and vertical angles are configurations that happen to guarantee a numerical fact, which is exactly what makes them useful in proofs.
Concept
Two lines, rays or segments are perpendicular when they intersect to form a right angle.
\[ \overleftrightarrow{AB} \perp \overleftrightarrow{CD} \]
One right angle is enough. If two lines cross and one of the four angles is 90, then its vertical angle is also 90 and each linear pair partner is 180 minus 90, so all four are right angles.
That short argument is a good first proof to be able to reproduce, because it uses vertical angles and linear pairs together and both were just established.
Socratic
Two lines cross and exactly one of the four angles is marked with a square.
Discussion prompt
Explain, using the two facts from this part, why the other three must also be 90 degrees.
Hint: Use the vertical angle relationship for one of them and the linear pair relationship for the other two.
Answer:
The angle opposite the marked one is vertical to it, and vertical angles are congruent, so it is also 90.
Each of the two remaining angles forms a linear pair with a 90 degree angle, so each measures 180 minus 90, which is 90.
So one square mark determines all four angles. This is why diagrams only ever mark one of them, and why perpendicularity is such a strong piece of given information in a problem.
Check
Solve it on paper before you click.
Check your understanding
Two lines intersect. One angle measures 4x plus 7 and the angle vertical to it measures 2x plus 33. What is the measure of each of those angles?
Answer: B
Why: Vertical angles are congruent, so the two expressions are equal. Solving 4x plus 7 equals 2x plus 33 gives 2x equals 26, so x is 13. Substituting back, 4 times 13 plus 7 is 59, and 2 times 13 plus 33 is also 59, confirming both the value and the setup.
Sorting
A diagram shows two lines crossing, with one angle marked 40 degrees. Sort what you may now use.
Sort into buckets
Sort each statement by whether the diagram guarantees it.
Notice the two failures are different kinds. One is contradicted by the given information, and one is simply unsupported. Both are equally unusable.
Elimination
Two lines cross. You know one angle is 65 and you want to claim the angle next to it is 115.
Eliminate the wrong options
Which reason correctly justifies that step?
Survives elimination: y2
Why: The two angles are adjacent with their outer sides forming a straight line, which is the definition of a linear pair, and a linear pair always totals 180. So the second angle is 180 minus 65, which is 115. The reason must name the configuration, not the appearance, and not a different relationship that happens to involve two angles.
Section
Closing the chapter with polygons, perimeter and area
Concept
A polygon is a closed figure made of line segments that meet only at their endpoints, with exactly two segments meeting at each vertex.
| sides | name |
|---|---|
| 3 | triangle |
| 4 | quadrilateral |
| 5 | pentagon |
| 6 | hexagon |
| 8 | octagon |
| n | n-gon |
Definition probe
Apply the three conditions rather than judging by appearance.
Sort into buckets
Sort each figure by whether it is a polygon.
Concept
Two more classifications, and they are independent of each other.
Convex — No side, when extended, passes through the interior. Equivalently, no interior angle is greater than 180 degrees.
Concave — At least one side, when extended, cuts through the interior. There is at least one interior angle greater than 180.
Regular — Convex, with all sides congruent and all angles congruent. A regular figure must be convex by definition.
A rhombus that is not a square has all sides congruent and is not regular, because its angles differ. Both conditions are required.
Counterexample
A student claims that a figure with all sides congruent must be regular.
Discussion prompt
Produce a counterexample, and say which of the two required conditions it breaks.
Hint: Think of a quadrilateral you can push over without changing its side lengths.
Answer:
A rhombus that is not a square. All four sides are congruent, and its angles come in two different sizes, so it fails the equal-angles condition.
The reverse counterexample also exists: a non-square rectangle has all angles congruent at 90 degrees and two different side lengths, so equal angles alone is not enough either.
Both conditions are genuinely needed, and the square is exactly the quadrilateral satisfying both. That is why regular quadrilateral and square mean the same thing.
Concept
Perimeter is the distance around a polygon, found by adding the side lengths. For a circle the same idea is called circumference.
\[ C = 2\pi r = \pi d \]
\[ A_{\text{circle}} = \pi r^2, \qquad A_{\text{triangle}} = \tfrac{1}{2}bh, \qquad A_{\text{rectangle}} = \ell w \]
Perimeter is measured in units of length and area in square units. Reporting an area without squaring the unit is a marked error, and it is also a useful self-check: if your working produced a length times a length, the answer is an area.
Worked example
This is where the first deck's distance formula returns. The triangle has vertices at A, B and C as given.
\[ A(1, 2), \qquad B(4, 6), \qquad C(1, 6) \]
Find each side length, choosing the easiest method for each.
Why: Two of these sides are vertical or horizontal, so their lengths are simple differences. Only the slanted side needs the distance formula, and noticing that saves most of the work.
\[ CA = 6 - 2 = 4, \qquad CB = 4 - 1 = 3 \]
Use the distance formula for the slanted side.
Why: This is the formula from the first deck, unchanged.
\[ AB = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Add the three sides for the perimeter.
Why: Perimeter is just the total distance around, so the three lengths are added.
\[ P = 4 + 3 + 5 = 12 \]
Use the two perpendicular sides as base and height for the area.
Why: The horizontal and vertical sides meet at a right angle at C, so they serve directly as the base and the height without any extra work.
\[ A = \tfrac{1}{2}(3)(4) = 6 \]
Figure (svg): A right triangle plotted on the coordinate plane with vertices at one comma two, four comma six and one comma six, with side lengths three, four and five marked.
Verify: check the side lengths satisfy the Pythagorean relationship.
Why: Three squared plus four squared is 9 plus 16, which is 25, and that equals five squared. The three lengths are consistent with a right triangle, which confirms both the distance calculation and the decision to use those two sides as base and height.
Estimation
A rectangle measures 8 units by 5 units.
Predict first
Which pair of values is correct?
Correct: perimeter 26, area 40
Why: Perimeter adds all four sides: 8 plus 5 plus 8 plus 5, which is 26 units. Area multiplies the two dimensions: 8 times 5, which is 40 square units. The first option has them swapped, which is the most common error, and the units are the giveaway since a product of two lengths must be the square-unit quantity.
Ranking
Four figures, all with an area of 36 square units. Order them from smallest perimeter to largest.
Put in order
Why: The circle has circumference about 21.3, the square has perimeter 24, the 4 by 9 rectangle has 26, and the 2 by 18 rectangle has 40. Same area, very different perimeters. The pattern is that the more elongated a figure is, the more perimeter it needs to enclose the same area, and the circle is the extreme case in the other direction.
Real world
The last exercise is not only a formula drill.
Discussion prompt
Where does the fact that a circle encloses the most area for a given perimeter actually show up?
Hint: Think about anything where the boundary costs money or energy.
Answer:
Fencing a field: for a fixed length of fence, the shape enclosing the most land is a circle, and among rectangles it is the square. Long thin plots waste fence.
Heat loss: a building loses heat through its surface, so compact shapes are cheaper to heat than sprawling ones with the same floor area. The same relationship, one dimension up.
It is also why bubbles are spherical and why cells are roughly round. Minimising boundary for a given content is a physical tendency, not only a geometric curiosity.
Check
Solve it on paper before you click.
Check your understanding
A triangle has vertices at P at zero comma zero, Q at six comma zero, and R at six comma eight. What are its perimeter and area?
Answer: A
Why: The horizontal side is 6 and the vertical side is 8, and they meet at a right angle at Q. The hypotenuse is the square root of 36 plus 64, which is the square root of 100, so 10. The perimeter is 6 plus 8 plus 10, which is 24, and the area is half of 6 times 8, which is 24. The two happening to be equal here is a coincidence worth noticing rather than a rule.
Trap
The rectangle is 8 by 5, so the area is 40 units. Perimeter and area are both just numbers describing the figure.
\[ A = 8 \times 5 = 40 \text{ units} \]
Area is measured in square units, not units, and writing the wrong one is treated as a wrong answer on most marking schemes rather than as a presentational slip.
\[ A = 8 \times 5 = 40 \text{ square units} \]
The reason it matters is that the unit records what kind of quantity you computed. A length times a length is an area, and the squared unit is the arithmetic saying so.
This gives you a free self-check. If your working multiplied two lengths, the answer must be in square units, and if it added lengths, it must be in plain units. An answer whose unit does not match its arithmetic has an error in one of the two.
Trade off
Fill the missing cells. Choosing the right formula is usually about which measurements you actually have.
Comparison matrix
| You are given | Reach for | Why |
|---|---|---|
| three vertices on the coordinate plane | distance formula for each side | you need lengths before any perimeter |
| a base and a perpendicular height | half base times height | the triangle area formula |
| a radius | two pi r, and pi r squared | circumference and area of a circle |
| a right angle between two known sides | use those two as base and height | no extra height calculation is needed |
The last row is the shortcut worth remembering. When a triangle has a right angle, the two sides forming it are already the base and the height, and hunting for an altitude is wasted work.
Connect it up
One page covering both halves of Chapter 1, which is also your revision sheet for the test.
Draw it
Divide a page into two columns, headed segments and angles. In the left column list segment measure, the segment addition postulate, congruent segments, midpoint and the distance formula. In the right column write the angle twin of each one, drawing a small diagram beside each. Underneath, add the four relationships that only exist for angles: linear pair, vertical, complementary and supplementary, with the equation each one produces. Finally, at the bottom, write the four polygon facts: what makes a polygon, convex against concave, regular, and the perimeter and area formulas.
The right column being almost a copy of the left is the point. Bring anything you cannot fill in and we will work it in the session.
Exit ticket
The idea that made this whole deck shorter than it looked.
Predict first
You are told ray BD bisects angle ABC. Which fact from the first deck is this the same as?
Correct: A midpoint dividing a segment into two congruent segments
Why: A bisector is the angle version of a midpoint: both cut a figure into two congruent halves, both let you set the two parts equal to each other, and both produce exactly the same kind of equation to solve. Recognising these parallels is what makes the second half of this chapter mostly review rather than new material.
Recap
Six things, and one parallel worth carrying into the next chapter.
| if the problem says | the equation is |
|---|---|
| a ray is drawn inside the angle | the two parts add to the whole |
| the ray bisects the angle | the two parts are equal to each other |
| the angles form a linear pair | the two measures total 180 |
| the angles are vertical | the two measures are equal |
| the angles are complementary | the two measures total 90 |
The parallel underneath everything: angles behave exactly like segments with degrees in place of length. Addition postulate, congruence, bisector and midpoint are the same four ideas told twice.
McGraw Hill, Reveal Geometry — Module 1, Tools of Geometry Module 1 — The chapter this completes.
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