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Triangle Angles and Theorems
Students use triangle angle relationships and major triangle theorems to find missing measures and justify geometric conclusions.
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Beginner
15–20 minutes
Triangles & Transformations
Triangles & Transformations
Page 4 of 8
Inside Cover
Triangle Angles and Theorems
At a Glance
Binder SectionTriangles & Transformations
DifficultyBeginner
Estimated Time15–20 minutes
PrerequisitesNone
How do angle and side relationships determine what triangles are possible?
Build Understanding
The interior angles of every triangle total 180 degrees. An exterior angle equals the sum of the two remote interior angles. These facts allow equations for unknown measures.
In an isosceles triangle, congruent sides have congruent opposite angles, and the converse is also true. An equilateral triangle is equiangular, with three 60-degree angles.
The Triangle Inequality Theorem says the sum of any two side lengths must be greater than the third. The longest side is opposite the largest angle.
A triangle midsegment connects the midpoints of two sides; it is parallel to the third side and half its length. Perpendicular bisectors, angle bisectors, medians, and altitudes create important points of concurrency.
Visual Model
4Thousands
3Hundreds
2Tens
7Ones
4,327 = 4,000 + 300 + 20 + 7
Goals
Learning Targets
- Use the 180-degree triangle sum.
- Apply the exterior angle theorem.
- Connect equal sides with equal opposite angles.
- Test side lengths with the triangle inequality.
- Use midsegment and concurrency relationships.
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Interactive Vocabulary
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Worked Examples
Example 1
Triangle angles are 48°, 67°, and x
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x=180−115=65°.
Example 2
An exterior angle is 124° and one remote interior angle is 53°
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Other remote angle is 71°.
Ask and explain
Common Questions
Can a triangle have two right angles? No.
Which angles are remote interior angles? The two nonadjacent to the exterior angle.
Does 2, 3, 5 form a triangle? No, because 2 + 3 is not greater than 5.
Are all equilateral triangles equiangular? Yes.
Notice and correct
Common Misconceptions
Subtracting an exterior angle from 180 without identifying the target
Mark adjacent and remote angles.
Pairing the largest angle with the shortest side
Larger angles face longer sides.
Using ≥ in triangle inequality
The sum must be strictly greater.
Learn Through Video
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Use the chapter markers, key vocabulary, and reflection prompts to stay actively engaged while you watch.
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Warm-Up
Guided
Independent
Challenge
1
Quick Start
Warm-Up
Find the third angle if two angles are 35° and 80°.
2
We Do
Guided Practice
Find the third angle if two angles are 48 and 67 degrees. Answer: 65 degrees.
An exterior angle is 125 degrees and one remote angle is 52 degrees. Answer: 73 degrees.
An isosceles triangle has vertex angle 40 degrees. Answer: each base angle is 70 degrees.
Can 4, 7, 12 form a triangle? Answer: no.
A midsegment corresponds to a side of length 18. Answer: midsegment = 9.
3
You Do
Independent Practice
Order sides opposite 40°, 60°, 80°.
Find x in angles x, 2x, and 3x.
Choose a possible third side with sides 7 and 12.
4
Stretch Your Thinking
Challenge & Real-World Practice
Create a triangle constraint puzzle using angle sum, exterior angle, and triangle inequality. Solve it and explain why no alternate triangle satisfies all clues.
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IXL
Find Missing Angles in Triangles
Open resourceExterior Angle Theorem
Open resourceTriangle Inequality
Open resourceMidsegments of Triangles
Open resourceKhan Academy
Triangles Unit
Open resourceTriangle Theorems
Open resourceDeltaMath
DeltaMath: Triangle Angle Sum
Open resourceDeltaMath: Exterior Angle Theorem
Open resourceDeltaMath: Isosceles Triangle Theorem
Open resourceDeltaMath: Triangle Inequality
Open resource
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Before you begin, I can…
- I use 180° for interior angle sum.
- I connect an exterior angle to its two remote interior angles.
- I test all required side-sum inequalities.
1
Question 1
Which statement shows the key idea in this lesson?
2
Question 2
Which habit best supports accuracy?
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Question 3
What should a student do after solving?
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Question 4
Which explanation is strongest?
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Question 5
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