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We are given two congruent triangles, \( \triangle BDC \cong \triangle SQR \), and need to find the following values:

### Corresponding sides and angles:
Since the triangles are congruent, corresponding sides and angles are equal.

From \( \triangle BDC \):
- Angle \( B = 63^\circ \)
- Angle \( D = 79^\circ \)
- Angle \( C = 38^\circ \)
- Side \( DC = 50 \text{ mi} \)

From \( \triangle SQR \):
- Angle \( S = 63^\circ \)
- Side \( SR = 55 \text{ mi} \)
- Side \( RQ = 50 \text{ mi} \)
- Side \( QS = 35 \text{ mi} \)

### Step 1: Finding \( CB \)
Since \( \triangle BDC \cong \triangle SQR \), the side \( CB \) corresponds to side \( SR \). Hence, 
\[ CB = SR = 55 \text{ miles}. \]

### Step 2: Finding \( m \angle LQ \)
Angle \( LQ \) corresponds to angle \( D \) in \( \triangle BDC \), which is given as:
\[ m\angle LQ = m\angle D = 79^\circ. \]

### Step 3: Finding \( m \angle LR \)
Angle \( LR \) corresponds to angle \( C \) in \( \triangle BDC \), which is given as:
\[ m\angle LR = m\angle C = 38^\circ. \]

### Step 4: Finding \( BD \)
Side \( BD \) corresponds to side \( SQ \) in \( \triangle SQR \), which is given as:
\[ BD = SQ = 35 \text{ miles}. \]

### Summary:
- \( CB = 55 \) miles
- \( m\angle LQ = 79^\circ \)
- \( m\angle LR = 38^\circ \)
- \( BD = 35 \) miles

Let me know if you want further explanations or details on any step.

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Here are 5 related questions:
1. What is the sum of interior angles in any triangle?
2. How do you prove that two triangles are congruent using the ASA (Angle-Side-Angle) criterion?
3. How can you find a missing side in a right triangle using the Pythagorean theorem?
4. What is the relation between corresponding sides of congruent triangles?
5. How can you solve for angles in a triangle if two angles are given?

**Tip:** In congruent triangles, corresponding angles and sides are equal.

The figures shown below are congruent such that ∆BDC ≅ ∆SQR. Find the values of CB, m∠Q, m∠R, and BD.

Solve for the sides and angles of congruent triangles ∆BDC and ∆SQR. Learn how to find the values of CB, m∠Q, m∠R, and BD using congruence properties. Ideal for students in Grades 8-10.

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