Geometry for RIT under 215: classifying triangles as acute, right or obtuse by their angles, as scalene, isosceles or equilateral by their sides, and giving a full two-word classification.
Subject: NWEA MAP Growth Math · 60 slides · symbolic lesson
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Title
MAP Growth · Geometry · RIT under 215
Naming a triangle by its angles, by its sides, and by both at once
Objectives
Three MAP skills sit here, and the key is that there are two separate naming systems.
The one idea: angles and sides give two independent names, so most triangles have a two-word description.
Section
Part 1
Concept
Every triangle can be described twice: once by its angles and once by its sides.
Figure (svg): A grid showing that every triangle has both an angle name and a side name
The two systems are independent, which is why a triangle can be a right scalene, or an acute isosceles.
Concept
This is the fact that makes every classification question answerable.
Figure (svg): A triangle with its three angles marked, summing to one hundred eighty degrees
Angle sum — The three angles of any triangle add to exactly 180 degrees, without exception.
Warm-up
Retrieve before being taught.
Discussion prompt
Name any types of triangle you can already remember, and say what makes each one that type.
Hint: Think about the triangles drawn in set squares and road signs.
Answer:
Most people remember right-angled and equilateral first.
Notice that those two come from different systems — one is about an angle and one is about the sides. That split is what this whole deck is about.
Pattern
Two questions, asked separately.
Asking them separately is what stops the two systems getting tangled.
Socratic
This explains why naming by the largest angle works at all.
Discussion prompt
Why is it impossible for a triangle to have two right angles, or two obtuse angles?
Hint: Add two 90-degree angles and see what is left.
Answer:
Because the three angles total 180 degrees. Two right angles already use all 180, leaving nothing for the third.
Two obtuse angles would exceed 180 on their own. So at most one angle can be 90 or more, which is why naming by the largest angle is unambiguous.
Definition probe
Sorting the vocabulary is the first step.
Sort into buckets
Sort each word by which system it belongs to.
Trap
Asked to classify a triangle by its sides, a student answers right-angled.
Right-angled is an angle name, not a side name. The side answer must be scalene, isosceles or equilateral.
Check which system the question is asking about. If it says by its sides, only three words are available.
Explain it to yourself
Being able to separate them is the whole skill.
Discussion prompt
In your own words, what is the difference between classifying by angles and classifying by sides?
Answer:
Classifying by angles looks at how wide the corners are, and reports the largest one.
Classifying by sides looks at how many edges have the same length. The two are measured differently and answered separately.
Analogy
Independent classifications are ordinary outside mathematics.
Match the pairs
Why: Knowing someone is tall tells you nothing about their hair, and knowing a triangle is right tells you almost nothing about its sides. Two independent properties need two separate answers.
Notation
The marks on a diagram carry the information, not the drawing.
Annotate
On: \( \text{tick marks} \;\Rightarrow\; \text{equal sides} \)
Read the marks before the shape — a diagram can be drawn misleadingly, but the marks cannot lie.
Section
Part 2
Concept
Since at most one angle can reach 90 degrees, the largest angle decides the name.
Figure (svg): Three triangles side by side: one acute, one right with a square mark, one obtuse
| name | largest angle | how to spot it |
|---|---|---|
| acute | less than 90 degrees | all three corners look sharp |
| right | exactly 90 degrees | a small square marks the corner |
| obtuse | more than 90 degrees | one corner is visibly spread open |
Worked example
A triangle has angles of 50, 60 and 70 degrees. Classify it by its angles.
Figure (svg): Three triangles side by side: one acute, one right with a square mark, one obtuse
Check the angle sum
Why: 50 plus 60 plus 70 is 180, so this is a valid triangle.
Find the largest angle
Why: The largest is 70 degrees.
Compare with 90
Why: 70 is less than 90, so every angle is acute.
\[ \text{largest} = 70^\circ < 90^\circ \;\Rightarrow\; \text{acute} \]
Verify: no angle reaches 90
Why: None of 50, 60 or 70 is 90 or more, so the triangle really is acute rather than right or obtuse.
Worked example
A triangle has angles of 35 and 55 degrees. Classify it by its angles.
Figure (svg): A triangle with its three angles marked, summing to one hundred eighty degrees
Use the angle sum
Why: The three angles total 180, so the third is 180 minus 35 minus 55.
Compute it
Why: 180 minus 90 is 90, so the third angle is 90 degrees.
Classify by the largest
Why: One angle is exactly 90, so this is a right triangle.
\[ 180 - 35 - 55 = 90 \;\Rightarrow\; \text{right} \]
Verify: the sum
Why: 35 plus 55 plus 90 is 180, which confirms the missing angle and therefore the classification.
Prediction
Find the third angle before deciding.
Predict first
A triangle has angles of 100 and 40 degrees. What type is it by its angles?
Correct: obtuse
You did not even need the third angle here — a given angle of 100 already exceeds 90.
Why: The third angle is 180 minus 100 minus 40, which is 40 degrees. The largest angle is 100, which is more than 90, so the triangle is obtuse.
Faded example
Two angles are known.
Fill in the blanks
180 - 25 - 65 = 90
Why: The three angles total 180, so the missing one is 90 degrees. That makes the triangle a right triangle.
Check
Solve it on paper before you click.
Check your understanding
A triangle has angles of 20, 40 and 120 degrees. What type is it?
Answer: A
Why: The largest angle is 120 degrees, which is more than 90, so the triangle is obtuse. The angles do total 180, so it is a valid triangle.
Elimination
Not every set of three angles can be a triangle.
Eliminate the wrong options
Which set of angles could form a triangle?
Survives elimination: e1
Why: The three angles must total exactly 180. Only 60, 60 and 60 does, and that triangle is both acute and equilateral.
Error analysis
A student classified a triangle with angles 30, 60 and 90.
Annotate
On: \( 30^\circ, 60^\circ, 90^\circ \;\Rightarrow\; \text{acute} \)
Always find the largest angle first — the smaller ones never decide the name.
Pattern
Each frame widens one angle while keeping the sum at 180.
Step through it
What happens to the other two angles as the largest one grows?
Because the sum is fixed at 180, one angle growing forces the others to shrink — which is why only one can ever exceed 90.
Invariant
One quantity never changes, whatever the shape.
Step through it
What is identical about all three?
The angle sum is invariant at 180 degrees. Every classification question ultimately rests on that one fixed fact.
Section
Part 3
Concept
The side classification is a straight count of how many sides share a length.
Figure (svg): Three triangles side by side: scalene, isosceles with two tick marks, equilateral with three
| name | equal sides | diagram mark |
|---|---|---|
| scalene | none | no tick marks |
| isosceles | two | matching ticks on two sides |
| equilateral | all three | matching ticks on all three |
Intuition
A diagram will rarely be drawn to scale, so the tick marks carry the information.
Figure (svg): Three triangles side by side: scalene, isosceles with two tick marks, equilateral with three
Two sides with the same number of ticks are equal, whatever they look like.
Worked example
A triangle has sides of 5 cm, 5 cm and 8 cm. Classify it by its sides.
Figure (svg): Three triangles side by side: scalene, isosceles with two tick marks, equilateral with three
Compare the lengths
Why: Two of them are 5 cm and one is 8 cm.
Count the equal sides
Why: Exactly two sides match.
Name it
Why: Two equal sides is isosceles.
\[ 5, 5, 8 \;\Rightarrow\; \text{isosceles} \]
Verify: it is not equilateral
Why: The third side is 8, not 5, so all three are not equal and the triangle is isosceles rather than equilateral.
Prediction
Count the matches.
Predict first
A triangle has sides 7 cm, 4 cm and 9 cm. What type is it by its sides?
Correct: scalene
Right is an angle name, so it could never be the answer to a by-sides question.
Why: All three lengths are different, so no sides are equal. A triangle with no equal sides is scalene.
Concept
The sides and angles are linked: the angles opposite equal sides are themselves equal.
Figure (svg): An equilateral triangle with all three angles marked as sixty degrees
So an equilateral triangle has three 60-degree angles, and an isosceles triangle has two equal angles.
Worked example
An isosceles triangle has an apex angle of 40 degrees. Find the other two angles.
Figure (svg): An equilateral triangle with all three angles marked as sixty degrees
Use the angle sum
Why: The other two angles together are 180 minus 40, which is 140.
Use the equal-sides link
Why: The two base angles are equal because the two sides are equal.
Split the remainder
Why: 140 divided by 2 is 70, so each base angle is 70 degrees.
\[ (180 - 40) \div 2 = 70^\circ \]
Verify: the sum and the type
Why: 40 plus 70 plus 70 is 180, and the largest angle is 70, so this is an acute isosceles triangle.
Faded example
The apex angle is 50 degrees.
Fill in the blanks
(180 - 50) \div 2 = 65
Why: The two base angles share the remaining 130 degrees equally, so each is 65 degrees. The triangle is acute isosceles.
Check
Solve it on paper before you click.
Check your understanding
A triangle has all three sides measuring 6 cm. What type is it by its sides?
Answer: A
Why: All three sides are equal, which is the definition of equilateral. Its three angles are all 60 degrees, so it is also acute.
Discrimination
Count the equal lengths in each.
Sort into buckets
Sort each set of side lengths.
Real world
The classification is not just vocabulary — it changes what a shape can do.
Discussion prompt
Why are equilateral and isosceles triangles used so often in bridges and roof trusses?
Answer:
Because a triangle cannot be pushed out of shape without changing a side length, which makes it the most rigid simple shape.
Equal sides spread load evenly, so equilateral and isosceles triangles distribute force symmetrically. A scalene triangle would load one member harder than the others.
Ranking
Count the matching lengths in each.
Put in order
Why: The first and last have no equal sides and are scalene, the second has two equal and is isosceles, and the third has all three equal and is equilateral. Ordering by equal-side count puts the two scalene triangles first.
Section
Part 4
Concept
A full classification gives one word from each system.
Figure (svg): A grid showing that every triangle has both an angle name and a side name
A triangle with a right angle and no equal sides is a right scalene triangle.
Worked example
A triangle has angles 90, 45 and 45 degrees, and two equal sides. Classify it fully.
Figure (svg): A grid showing that every triangle has both an angle name and a side name
Classify by angles
Why: The largest angle is exactly 90, so it is a right triangle.
Classify by sides
Why: Two sides are equal, so it is isosceles.
Join the two names
Why: A right isosceles triangle.
\[ 90^\circ, 45^\circ, 45^\circ \;\Rightarrow\; \text{right isosceles} \]
Verify: the two equal angles
Why: The two 45-degree angles are equal, which matches the two equal sides — the two classifications agree with each other.
Intuition
Most combinations exist, but a few are impossible.
Figure (svg): An equilateral triangle with all three angles marked as sixty degrees
Prediction
Think about what equal sides force.
Predict first
Can a triangle be both right-angled and equilateral?
Correct: no, because equilateral forces three 60 degree angles
This is the one place the two systems are not fully independent.
Why: Equal sides force equal angles, and three equal angles totalling 180 must each be 60 degrees. Since 60 is not 90, an equilateral triangle can never contain a right angle.
Comparison
Each row is one triangle described both ways.
Comparison matrix
| angles | sides | full name |
|---|---|---|
| 90, 60, 30 | all different | right scalene |
| 90, 45, 45 | two equal | right isosceles |
| 60, 60, 60 | all equal | acute equilateral |
| 100, 40, 40 | two equal | obtuse isosceles |
Notice that the angle word and the side word are chosen completely separately in every row.
Sorting
Most can, but equilateral is restrictive.
Sort into buckets
Sort each described triangle by whether it can exist.
Equilateral is the only side type that fixes the angles, and it fixes them at 60 degrees each.
Check
Solve it on paper before you click.
Check your understanding
A triangle has angles of 110, 35 and 35 degrees. What is its full classification?
Answer: A
Why: The largest angle is 110, which is over 90, so it is obtuse. Two angles are equal at 35 degrees, which means two sides are equal, so it is isosceles.
Error analysis
A student classified a triangle with sides 5, 5, 5.
Annotate
On: \( 5, 5, 5 \;\Rightarrow\; \text{right equilateral} \)
Whenever you see equilateral, the angle classification is already decided for you.
Section
Part 5
Concept
Every classification item yields to the same two questions asked separately.
Figure (svg): A grid showing that every triangle has both an angle name and a side name
Largest angle for the first word, count of equal sides for the second.
Step zero
One check settles which system is being asked about.
Discussion prompt
You are asked to classify a triangle. What should you check about the question itself first?
Hint: The check is about the wording, not the diagram.
Answer:
Whether it asks by angles, by sides, or for a full classification.
Answering with the wrong system loses the mark even when the observation is correct — a right triangle is not an answer to a by-sides question.
Matching
Use both systems on each.
Match the pairs
Why: Equal angles mean equal sides, so counting repeated angles gives the side classification without any lengths being stated at all.
Two truths and a lie
Two of these are sound; one is not.
Eliminate the wrong options
Which statement about triangles is false?
Survives elimination: t2
Why: Two right angles would use all 180 degrees between them, leaving nothing for the third angle. At most one angle in a triangle can be 90 degrees or more.
Ranking
Classify each, then order.
Put in order
Why: The largest angles are 60, 80, 90 and 130, so the triangles run acute, acute, right, obtuse. The angle classification follows the same order.
Explain it
A classmate answers isosceles when asked to classify by angles.
Discussion prompt
How would you make the split between the two systems stick?
Answer:
Give them two lists side by side — acute, right, obtuse in one and scalene, isosceles, equilateral in the other — and have them label which is measured with a protractor and which with a ruler.
Tying each system to a different measuring tool makes the split physical rather than verbal.
Missing information
Not every question can be answered as asked.
Discussion prompt
A question gives a triangle with one angle of 40 degrees and asks for its full classification. What is missing?
Answer:
At least one more angle, or some information about the sides.
One angle of 40 leaves 140 to share between the other two, which could be 70 and 70, or 100 and 40, or many other splits. Those give different classifications.
Estimation
Comparing against a right angle is the practical method on a diagram.
Predict first
A triangle has one corner that is visibly wider than the corner of a page. What type is it?
Correct: obtuse
Only one angle can be over 90, so spotting one settles the classification immediately.
Why: The corner of a page is a right angle of 90 degrees. An angle visibly wider than that is more than 90, which makes the triangle obtuse.
Edge cases
Test what happens at the extreme.
Discussion prompt
How large can one angle of a triangle get, and what happens as it approaches that limit?
Hint: Think about what the other two angles are doing.
Answer:
It can approach 180 degrees but never reach it, because the other two angles must share what is left and each must be more than zero.
As one angle approaches 180 the triangle flattens towards a straight line. That is why 180 is excluded — a triangle with a 180-degree angle would not be a triangle at all.
Commit first
Decide your answer and your confidence before revealing.
Predict first
A triangle has two angles of 45 degrees. What is its full classification?
Correct: right isosceles
Two equal angles always mean two equal sides, so the side classification came free.
Why: The third angle is 180 minus 45 minus 45, which is 90, so it is a right triangle. The two equal angles force two equal sides, so it is isosceles.
Exit ticket
One item that tells you whether the deck landed.
Predict first
A triangle has sides 8, 8, 8 and you are asked to classify it by its angles. What is the answer?
Correct: acute
Why: Equal sides force equal angles, so all three are 60 degrees. Since every angle is under 90, the triangle is acute. Equilateral would answer a by-sides question, not this one.
Counterexample
A rule is worth more once you have tried to break it.
Discussion prompt
A student claims every isosceles triangle is acute. Find a counterexample.
Answer:
A triangle with angles 120, 30 and 30 has two equal angles, so two equal sides, making it isosceles — and its largest angle is 120, so it is obtuse.
Isosceles constrains only the sides, so it leaves the angle classification completely open. Equilateral is the only side type that fixes the angles.
Connect it up
Your handwriting, one page, from memory.
Draw it
Draw three triangles classified by angle with the largest angle marked in each, and three classified by side with tick marks. Beside them write the two questions you ask, and note the one impossible combination.
If you can draw this from memory, all three skills in this deck are covered.
Recap
Three skills, two systems.
Read whether the question wants angles, sides or both — answering with the right observation in the wrong system still loses the mark.
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