Grade: Grade 10 Subject: Mathematics Unit: Geometry Theorems Lesson: 1 of 6 SAT: Geometry+Trigonometry ACT: Math

Triangle Theorems

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Triangle theorems form the foundation of geometry. Understanding these theorems helps you solve problems involving angles, sides, and relationships within triangles.

Triangle Sum Theorem

The sum of the interior angles of any triangle is exactly 180°.

If triangle ABC has angles A, B, and C, then: m∠A + m∠B + m∠C = 180°

Exterior Angle Theorem

The measure of an exterior angle of a triangle equals the sum of the two remote interior angles (the two angles not adjacent to the exterior angle).

If angle D is an exterior angle at vertex C, then: m∠D = m∠A + m∠B

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

For triangle with sides a, b, c:

  • a + b > c
  • a + c > b
  • b + c > a

This helps determine if three given lengths can form a triangle.

Isosceles Triangle Theorems

Theorem Statement
Base Angles Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Converse If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Triangle Congruence Theorems

Theorem Meaning What You Need
SSS Side-Side-Side All three pairs of corresponding sides are congruent
SAS Side-Angle-Side Two sides and the included angle are congruent
ASA Angle-Side-Angle Two angles and the included side are congruent
AAS Angle-Angle-Side Two angles and a non-included side are congruent
HL Hypotenuse-Leg Hypotenuse and one leg of right triangles are congruent
Warning: AAA (Angle-Angle-Angle) does NOT prove congruence - it only proves similarity. SSA (Side-Side-Angle) is also not a valid congruence theorem (it can produce two different triangles).

Triangle Similarity Theorems

Theorem Meaning
AA Two pairs of corresponding angles are congruent
SSS Similarity All three pairs of corresponding sides are proportional
SAS Similarity Two pairs of sides are proportional AND the included angles are congruent
SAT/ACT Connection: Triangle theorems appear frequently on both tests. The Triangle Sum Theorem and Exterior Angle Theorem are especially common. You'll also need to recognize when triangles are similar and set up proportions to find missing side lengths.

💡 Examples

Example 1: Using the Triangle Sum Theorem

Problem: In triangle ABC, m∠A = 45° and m∠B = 72°. Find m∠C.

Solution:

By the Triangle Sum Theorem: m∠A + m∠B + m∠C = 180°

45° + 72° + m∠C = 180°

117° + m∠C = 180°

m∠C = 180° - 117°

Answer: m∠C = 63°

Example 2: Using the Exterior Angle Theorem

Problem: In triangle PQR, an exterior angle at R measures 130°. If m∠P = 55°, find m∠Q.

Solution:

By the Exterior Angle Theorem: exterior angle = sum of remote interior angles

130° = m∠P + m∠Q

130° = 55° + m∠Q

m∠Q = 130° - 55°

Answer: m∠Q = 75°

Example 3: Triangle Inequality

Problem: Can a triangle have sides of length 5, 8, and 15?

Solution:

Check the Triangle Inequality Theorem:

5 + 8 = 13. Is 13 > 15? No, 13 < 15.

Since 5 + 8 is NOT greater than 15, these lengths cannot form a triangle.

Answer: No, a triangle cannot have these side lengths.

Example 4: Isosceles Triangle

Problem: In isosceles triangle XYZ, XY = XZ and m∠Y = 65°. Find m∠X and m∠Z.

Solution:

Step 1: By the Base Angles Theorem, since XY = XZ, the angles opposite these sides are equal.

∠Y and ∠Z are the base angles, so m∠Z = m∠Y = 65°

Step 2: Use the Triangle Sum Theorem

m∠X + m∠Y + m∠Z = 180°

m∠X + 65° + 65° = 180°

m∠X = 180° - 130° = 50°

Answer: m∠X = 50° and m∠Z = 65°

Example 5: Similar Triangles and Proportions

Problem: Triangles ABC and DEF are similar with AB/DE = 3/5. If AB = 9 and BC = 12, find DE and EF.

Solution:

Step 1: Find the scale factor

AB/DE = 3/5, and AB = 9

9/DE = 3/5

DE = 9 × (5/3) = 15

Step 2: Find EF using the same ratio

BC/EF = 3/5

12/EF = 3/5

EF = 12 × (5/3) = 20

Answer: DE = 15 and EF = 20

✏️ Practice

Try these problems on your own. Choose the best answer for each question.

1. In a triangle, two angles measure 38° and 94°. What is the measure of the third angle?

A) 42°

B) 48°

C) 52°

D) 132°

2. An exterior angle of a triangle measures 115°. One of the remote interior angles is 47°. What is the other remote interior angle?

A) 68°

B) 65°

C) 62°

D) 47°

3. Which set of lengths CAN form a triangle?

A) 2, 3, 6

B) 4, 4, 8

C) 5, 7, 11

D) 1, 2, 4

4. In an isosceles triangle, the vertex angle measures 40°. What is the measure of each base angle?

A) 40°

B) 70°

C) 80°

D) 140°

5. Which theorem would you use to prove two triangles congruent if you know two angles and the included side?

A) SSS

B) SAS

C) ASA

D) AAS

6. Triangle ABC is similar to triangle DEF. If AB = 6, BC = 8, and DE = 9, what is EF?

A) 10

B) 12

C) 14

D) 16

7. In a triangle, the angles are in the ratio 2:3:4. What is the largest angle?

A) 40°

B) 60°

C) 80°

D) 100°

8. Which is NOT a valid way to prove triangles congruent?

A) SSS

B) SSA

C) ASA

D) HL

9. The sides of a triangle are 7, 10, and x. Which is a possible value of x?

A) 2

B) 3

C) 17

D) 10

10. In an equilateral triangle, what is the measure of each angle?

A) 45°

B) 60°

C) 90°

D) 120°

Click to reveal answers
  1. B) 48° — 180° - 38° - 94° = 48°
  2. A) 68° — 115° - 47° = 68°
  3. C) 5, 7, 11 — 5 + 7 = 12 > 11, 5 + 11 = 16 > 7, 7 + 11 = 18 > 5 (all three conditions met)
  4. B) 70° — Base angles are equal: (180° - 40°) ÷ 2 = 70°
  5. C) ASA — Angle-Side-Angle where the side is between the two angles
  6. B) 12 — Scale factor is 9/6 = 3/2, so EF = 8 × (3/2) = 12
  7. C) 80° — 2x + 3x + 4x = 180°, so 9x = 180°, x = 20°. Largest = 4(20°) = 80°
  8. B) SSA — SSA (Side-Side-Angle) is ambiguous and not valid for congruence
  9. D) 10 — x must satisfy: 7 + 10 > x, 7 + x > 10, 10 + x > 7, giving 3 < x < 17
  10. B) 60° — All angles equal: 180° ÷ 3 = 60°

✅ Check Your Understanding

Question 1: If you know two angles of a triangle, can you always find the third? Why?

Reveal Answer

Yes, you can always find the third angle because the Triangle Sum Theorem guarantees that the three interior angles of any triangle sum to exactly 180°. If you know two angles, the third angle is simply 180° minus the sum of the known angles. This is one of the most fundamental and useful properties in geometry.

Question 2: What is the relationship between an exterior angle and the interior angle at the same vertex?

Reveal Answer

An exterior angle and its adjacent interior angle are supplementary, meaning they add up to 180°. This is because they form a linear pair (they share a side and their other sides form a straight line). For example, if an interior angle is 70°, the exterior angle at that vertex is 180° - 70° = 110°. This relationship connects the Exterior Angle Theorem to the Triangle Sum Theorem.

Question 3: Why does AAA (knowing all three angles are equal) prove similarity but not congruence?

Reveal Answer

AAA proves similarity because equal angles guarantee the same shape. However, it doesn't prove congruence because triangles can have identical angles but different sizes. Think of two equilateral triangles - one small and one large. Both have three 60° angles (same shape), but they're not congruent (different size). To prove congruence, you need information about at least one side to establish the actual size of the triangle.

Question 4: Given two sides of a triangle are 6 and 10, what is the range of possible values for the third side?

Reveal Answer

Let the third side be x. By the Triangle Inequality Theorem: (1) 6 + 10 > x, so x < 16; (2) 6 + x > 10, so x > 4; (3) 10 + x > 6, which is always true when x > 0. Combining these, the third side must be greater than 4 and less than 16. In interval notation: 4 < x < 16. So x can be any value between 4 and 16, not including 4 or 16.

🚀 Next Steps

  • Review any concepts that felt challenging
  • Move on to the next lesson when ready
  • Return to practice problems periodically for review