**Problem 1 : **

Explain whether the triangles PQR and STU are similar.

**Problem 2 : **

Explain whether the triangles ABC and DEF are similar.

**Problem 3 : **

Explain whether the triangles ABC and DEF are similar.

**Problem 1 : **

Explain whether the triangles PQR and STU are similar.

**Solution :**

The figure shows only one pair of congruent angles. Find the measure of the third angle in each triangle.

**Triangle PQR : **

Write the Triangle Sum Theorem for this triangle.

m∠P + m∠Q + m∠R = 180°

Substitute the given angle measures.

45° + 100° + m∠R = 180°

Simplify.

145° + m∠R = 180°

Subtract 145° from both sides.

145° + m∠R - 145° = 180° - 145°

Simplify.

m∠R = 35°

**Triangle STU : **

Write the Triangle Sum Theorem for this triangle.

m∠S + m∠T + m∠U = 180°

Substitute the given angle measures.

m∠S + 100° + 35° = 180°

Simplify.

m∠S + 135° = 180°

Subtract 135° from both sides.

m∠S + 135° - 135° = 180° - 135°

Simplify.

m∠S = 45°

**Conclusion : **

Three Angles of triangle PQR are 45°, 100° and 35°.

Three Angles of triangle STU are 45°, 100° and 35°.

Because two angles in one triangle are congruent to two angles in the other triangle, the two triangles are similar.

**Problem 2 : **

Explain whether the triangles ABC and DEF are similar.

**Solution :**

The figure shows only one pair of congruent angles. Find the measure of the third angle in each triangle.

**Triangle ABC : **

Write the Triangle Sum Theorem for this triangle.

m∠A + m∠B + m∠C = 180°

Substitute the given angle measures.

m∠A + 58° + 70° = 180°

Simplify.

m∠A + 128° = 180°

Subtract 128° from both sides.

m∠A + 128° - 128° = 180° - 128°

Simplify.

m∠A = 52°

**Triangle DEF : **

Write the Triangle Sum Theorem for this triangle.

m∠D + m∠E + m∠F = 180°

Substitute the given angle measures.

70° + m∠E + 49° = 180°

Simplify.

m∠E + 119° = 180°

Subtract 119° from both sides.

m∠E + 119° - 119° = 180° - 119°

Simplify.

m∠E = 61°

**Conclusion :**

Three Angles of triangle ABC are 52°, 58° and 70°.

Three Angles of triangle DEF are 70°, 61° and 49°.

Because only one angle is congruent, the two triangles are not similar.

**Problem 3 : **

Explain whether the triangles ABC and DEF are similar.

**Solution :**

The figure shows only one pair of congruent angles. Find the measure of the third angle in each triangle.

**Triangle ABC : **

Write the Triangle Sum Theorem for this triangle.

m∠A + m∠B + m∠C = 180°

Substitute the given angle measures.

41° + m∠B + 30° = 180°

Simplify.

m∠B + 71° = 180°

Subtract 71° from both sides.

m∠B + 71° - 71° = 180° - 71°

Simplify.

m∠B = 109°

**Triangle DEF : **

Write the Triangle Sum Theorem for this triangle.

m∠D + m∠E + m∠F = 180°

Substitute the given angle measures.

m∠D + 109° + 30° = 180°

Simplify.

m∠D + 139° = 180°

Subtract 130° from both sides.

m∠D + 139° - 139° = 180° - 139°

Simplify.

m∠D = 41°

**Conclusion :**

Three Angles of triangle ABC are 41°, 109° and 30°.

Three Angles of triangle DEF are 41°, 109° and 30°.

Because two angles in one triangle are congruent to two angles in the other triangle, the two triangles are similar.

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