Question

directions: determine whether the triangles are similar by angle - angle similarity. if yes, write a similarity statement.
1.
2.
3.
4.
5.
6.

Explanation:

Step1: Recall AA Similarity Criterion

Two triangles are similar by AA (Angle - Angle) similarity if two pairs of corresponding angles are congruent.

Step2: Analyze Problem 1

• $\angle CED=\angle GEF$ (vertical angles are congruent).
• $\angle D=\angle A$ (given as indicated by the arrows).
• So, $\triangle CDE\sim\triangle GAF$ by AA similarity.

Step3: Analyze Problem 2

• $\angle LQP=\angle LMN$ (corresponding angles as $QP\parallel MN$).
• $\angle L=\angle L$ (common angle).
• So, $\triangle LQP\sim\triangle LMN$ by AA similarity.

Step4: Analyze Problem 3

• In $\triangle SEY$, $\angle S = 180^{\circ}-\angle Y-\angle E=180^{\circ}-90^{\circ}-39^{\circ} = 51^{\circ}$.
• In $\triangle CHW$, $\angle C = 180^{\circ}-\angle H-\angle W=180^{\circ}-90^{\circ}-51^{\circ}=39^{\circ}$.
• $\angle S=\angle W = 51^{\circ}$ and $\angle E=\angle C = 39^{\circ}$.
• So, $\triangle SEY\sim\triangle CHW$ by AA similarity.

Step5: Analyze Problem 4

• In $\triangle ABD$, $\angle B=180^{\circ}-\angle A-\angle D=180^{\circ}-68^{\circ}-54^{\circ}=58^{\circ}$.
• In $\triangle EGF$, $\angle E=180^{\circ}-\angle G-\angle F=180^{\circ}-63^{\circ}-68^{\circ}=49^{\circ}$.
• No two pairs of congruent angles. So, not similar by AA.

Step6: Analyze Problem 5

• In $\triangle LMN$, $\angle M=180^{\circ}-\angle L-\angle N=180^{\circ}-25^{\circ}-67^{\circ}=88^{\circ}$.
• In $\triangle LBC$, $\angle LBC = 180^{\circ}-\angle L-\angle BCL=180^{\circ}-25^{\circ}-91^{\circ}=64^{\circ}$.
• No two pairs of congruent angles. So, not similar by AA.

Step7: Analyze Problem 6

• $\angle PXQ=\angle YXZ$ (vertical angles are congruent).
• If we assume no other angle - congruence information (from the given figure, no other angles are marked as equal). So, not enough information to prove similarity by AA.

Answer:

1. $\triangle CDE\sim\triangle GAF$
2. $\triangle LQP\sim\triangle LMN$
3. $\triangle SEY\sim\triangle CHW$
4. Not similar by AA
5. Not similar by AA
6. Not enough information to prove similarity by AA