https://file.aiquickdraw.com/mathbot/file/172730181985440hkt4tm.jpg

Let's analyze the figure step by step to find the measure of ∠RCD.

### Key Observations:
1. Triangle **PDC** has:
   - \( \angle PCD = 35^\circ \)
   - \( PD = CD = 11 \) (making triangle PDC isosceles)

2. Triangle **CRD** has:
   - \( CD = CR = 11 \) (since both sides are marked with the same length, making triangle CRD isosceles)

We are asked to find \( \angle RCD \).

### Steps:
1. **Using Isosceles Triangle Theorem in ∆PDC**:
   - Since \( PD = CD \), the base angles \( \angle PCD \) and \( \angle PDC \) are congruent.
   - So, \( \angle PDC = \angle PCD = 35^\circ \).

2. **Using the Sum of Angles in ∆PDC**:
   - The sum of the angles in a triangle is always \( 180^\circ \).
   - Thus, \( \angle P + 2 \times 35^\circ = 180^\circ \).
   - This gives us \( \angle P = 180^\circ - 70^\circ = 110^\circ \).

3. **Using Isosceles Triangle Theorem in ∆CRD**:
   - Since \( CR = CD \), \( \angle RCD = \angle CRD \).

4. **Sum of Angles in ∆CRD**:
   - The sum of angles in triangle CRD is also \( 180^\circ \).
   - Let \( \angle RCD = x \). Thus, \( 2x + \angle DCR = 180^\circ \).
   - From the figure, \( \angle DCR = 35^\circ \).
   - Substituting this into the equation: \( 2x + 35^\circ = 180^\circ \).
   - Solving for \( x \): \( 2x = 145^\circ \), so \( x = 72.5^\circ \).

Therefore, the measure of \( \angle RCD \) is approximately **72.5°**, but the closest available answer is **70°**.

Thus, the correct answer is:
\[
\boxed{E. \, 70^\circ}
\]

Let me know if you would like a more detailed explanation!

---

### Relative Questions:
1. What is the sum of angles in any triangle?
2. How does the isosceles triangle property help in solving this problem?
3. What are the base angles in an isosceles triangle?
4. Can you explain how to calculate the remaining angle if two angles are known in a triangle?
5. What is the difference between acute and obtuse angles?

### Tip:
Always remember to check for isosceles triangles in geometric figures as they give key angle relationships!

What is the measure of ∠RCD in the figure below?

Discover how to find the measure of ∠RCD in a geometry problem involving isosceles triangles. This solution includes key concepts such as the angle sum of a triangle and the isosceles triangle theorem. Suitable for Grades 8-10 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation