Question

example 1
determine whether each pair of triangles is similar. explain your reasoning.
1.
2.
3.
4.

Explanation:

1. Step1: Identify congruent angles

$\angle F = \angle J$ (given), $\angle GHF = \angle JHK$ (vertical angles)

1. Step2: Apply AA Similarity Criterion

Two pairs of congruent angles mean $\triangle GFH \sim \triangle KJH$
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2. Step1: Calculate missing angles

For $\triangle STR$: $\angle R = 180^\circ - 42^\circ - 98^\circ = 40^\circ$
For $\triangle UVW$: $\angle W = 180^\circ - 40^\circ - 98^\circ = 42^\circ$

2. Step2: Match congruent angles

$\angle S = \angle W = 42^\circ$, $\angle T = \angle V = 98^\circ$, $\angle R = \angle U = 40^\circ$

2. Step3: Apply AA Similarity Criterion

Three pairs of congruent angles mean $\triangle STR \sim \triangle WVU$
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3. Step1: Identify congruent angles

$\angle BAC = \angle FDC$ (alternate interior angles), $\angle ABC = \angle DFC$ (alternate interior angles)

3. Step2: Apply AA Similarity Criterion

Two pairs of congruent angles mean $\triangle ABC \sim \triangle DFC$
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4. Step1: Calculate missing angles

For $\triangle SKM$: $\angle M = 180^\circ - 90^\circ - 60^\circ = 30^\circ$
For $\triangle QRT$: $\angle T = 180^\circ - 90^\circ - 30^\circ = 60^\circ$

4. Step2: Match congruent angles

$\angle S = \angle R = 90^\circ$, $\angle K = \angle T = 60^\circ$, $\angle M = \angle Q = 30^\circ$

4. Step3: Apply AA Similarity Criterion

Three pairs of congruent angles mean $\triangle SKM \sim \triangle RTQ$

Answer:

1. $\triangle GFH \sim \triangle KJH$ by AA Similarity (two pairs of congruent angles: $\angle F \cong \angle J$, vertical angles $\angle GHF \cong \angle JHK$)
2. $\triangle STR \sim \triangle WVU$ by AA Similarity (all corresponding angles are congruent: $\angle S=42^\circ=\angle W$, $\angle T=98^\circ=\angle V$, $\angle R=40^\circ=\angle U$)
3. $\triangle ABC \sim \triangle DFC$ by AA Similarity (two pairs of congruent alternate interior angles: $\angle BAC \cong \angle FDC$, $\angle ABC \cong \angle DFC$)
4. $\triangle SKM \sim \triangle RTQ$ by AA Similarity (all corresponding angles are congruent: $\angle S=90^\circ=\angle R$, $\angle K=60^\circ=\angle T$, $\angle M=30^\circ=\angle Q$)