In the diagram below,   start overline, A, B, end overline, \parallel, start overline, D, F, end overline, comma 
AB
 ∥ 
DF
 ,   m, angle, D, C, E, equals, 105, degreesm∠DCE=105 
∘
  and m, angle, C, E, F, equals, 150, degrees, .m∠CEF=150 
∘
 . Find m, angle, B, .m∠B.


To solve for \( m\angle B \), let's break down the problem step by step.

### Given:
- \( \overline{AB} \parallel \overline{DF} \) (AB is parallel to DF),
- \( m\angle DCE = 105^\circ \),
- \( m\angle CEF = 150^\circ \).

Since \( \overline{AB} \parallel \overline{DF} \), we know that the angles involving these lines might be related through corresponding angles, alternate interior angles, or by applying other geometric properties.

#### Step 1: Analyze \( \triangle CEF \)
From the given, we have \( m\angle CEF = 150^\circ \). In a triangle, the sum of the internal angles is always 180 degrees. So, using the fact that the triangle has an angle of 150 degrees at \( E \), the other two angles must sum to \( 30^\circ \).

Let's first look at the relationship between angles formed by the parallel lines:

#### Step 2: Use parallel line properties
Since \( \overline{AB} \parallel \overline{DF} \) and we're dealing with a transversal through \( \overline{AB} \), you can use alternate interior or corresponding angles to relate \( \angle B \) to the angles on line DF. Based on the geometry setup, \( m\angle B \) is complementary to one of these angles.

#### Solution:
Now, apply these findings to the problem:

Let me know if you need any further details or clarification about the steps!

---

Here are 5 questions to expand on this topic:

1. How can you use corresponding angles to solve parallel line problems?
2. What are the key angle properties of a triangle?
3. How do you solve for an unknown angle in a transversal crossing parallel lines?
4. How do alternate interior angles help in parallel line problems?
5. How do supplementary and complementary angles differ in parallel line scenarios?

**Tip:** In any geometry problem, always look for parallel lines, as they often unlock key angle relationships like corresponding and alternate interior angles!

In the diagram below, AB ∥ DF, m∠DCE = 105°, and m∠CEF = 150°. Find m∠B.

This geometry problem involves using properties of parallel lines and the sum of internal angles in a triangle to find the measure of angle B. With AB ∥ DF, m∠DCE = 105°, and m∠CEF = 150°, explore how corresponding and alternate interior angles help solve the problem. Suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation