Question

07 - proving triangles similar

1. in the diagram below of δcer, la || cr. prove δcer ~ δlea.

statements | reasons
--- | ---

1. in the diagram below of δcer, la || cr. prove δcer ~ δlea | 1)
2. ∠e ≅ ∠e | 2)
3. | 3)
4. δcer ~ δlea | 4)

1. in the diagram below, ut || wx. prove δuvt ~ δxvw.

statements | reasons
--- | ---

1. in the diagram below, ut || wx, | 1)
2. ∠u ≅ ∠x | 2)
3. | 3) vertical angles are congruent.
4. δuvt ~ δxvw | 4)

Explanation:

Problem 1: Prove $\triangle CER \sim \triangle LEA$

Step 1: Identify Given

The problem states $LA \parallel CR$ (given in the diagram setup).

Step 2: Identify Common Angle

$\angle E \cong \angle E$ (reflexive property of congruence, as it is the same angle in both triangles).

Step 3: Identify Corresponding Angles

Since $LA \parallel CR$, $\angle ELA \cong \angle ECR$ (corresponding angles formed by transversal $EC$) and $\angle EAL \cong \angle ERC$ (corresponding angles formed by transversal $ER$). Alternatively, using one pair: $\angle ELA \cong \angle ECR$ (corresponding angles, $LA \parallel CR$).

Step 4: Apply AA Similarity

If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar (AA Similarity Criterion).

Filling the Table for Problem 1:

Statements	Reasons
2) $\angle E \cong \angle E$	2) Reflexive Property of Congruence
3) $\angle ELA \cong \angle ECR$ (or $\angle EAL \cong \angle ERC$)	3) Corresponding Angles Postulate (since $LA \parallel CR$)
4) $\triangle CER \sim \triangle LEA$	4) AA (Angle-Angle) Similarity Criterion

Problem 2: Prove $\triangle UVT \sim \triangle XVW$

Step 1: Identify Given

The problem states $\overline{UT} \parallel \overline{WX}$ (given).

Step 2: Identify Alternate Interior Angles

Since $UT \parallel WX$, $\angle U \cong \angle X$ (alternate interior angles formed by transversal $UX$).

Step 3: Identify Vertical Angles

$\angle UVT \cong \angle XVW$ (vertical angles are congruent, as given in the table’s reason for step 3).

Step 4: Apply AA Similarity

If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar (AA Similarity Criterion).

Filling the Table for Problem 2:

Statements	Reasons
2) $\angle U \cong \angle X$	2) Alternate Interior Angles Theorem (since $UT \parallel WX$)
3) $\angle UVT \cong \angle XVW$	3) Vertical angles are congruent.
4) $\triangle UVT \sim \triangle XVW$	4) AA (Angle-Angle) Similarity Criterion

Answer:

s (Table Completion):

Problem 1 Table:

1. Given
2. Reflexive Property of Congruence
3. $\angle ELA \cong \angle ECR$ (Corresponding Angles Postulate)
4. AA Similarity Criterion

Problem 2 Table:

1. Given
2. Alternate Interior Angles Theorem
3. $\angle UVT \cong \angle XVW$
4. AA Similarity Criterion