Math Problem Statement

∠C and ∠D are a linear pair. ∠C is four times larger than ∠D. What is the measure of ∠C?

Solution

To solve this problem, let's first break down the information provided:

  1. C\angle C and D\angle D are a linear pair. This means that they are adjacent angles formed by two intersecting lines, and their measures add up to 180180^\circ.

  2. C\angle C is four times larger than D\angle D. Mathematically, this can be written as: C=4×D\angle C = 4 \times \angle D

Let D=x\angle D = x. Then, C=4x\angle C = 4x.

Since C\angle C and D\angle D form a linear pair, we have: C+D=180\angle C + \angle D = 180^\circ Substituting the expressions for C\angle C and D\angle D into this equation: 4x+x=1804x + x = 180^\circ 5x=1805x = 180^\circ x=1805=36x = \frac{180^\circ}{5} = 36^\circ

Therefore, D=36\angle D = 36^\circ.

Now, to find C\angle C: C=4×36=144\angle C = 4 \times 36^\circ = 144^\circ

Final Answer:

The measure of C\angle C is 144144^\circ.

Would you like further details or explanations on this solution?

Here are five related questions to explore:

  1. What is the measure of D\angle D when C\angle C and D\angle D are linear pairs and C\angle C is twice D\angle D?
  2. Can the measure of C\angle C ever be greater than 180180^\circ if C\angle C and D\angle D are linear pairs?
  3. How do the properties of supplementary angles relate to linear pairs?
  4. What would happen if C\angle C was only three times D\angle D?
  5. How does the concept of vertical angles relate to linear pairs?

Tip: When dealing with linear pairs, remember that the sum of the angles will always be 180180^\circ. This property is key to solving many geometric problems involving straight lines and adjacent angles.

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Math Problem Analysis

Mathematical Concepts

Geometry
Linear Pairs
Supplementary Angles

Formulas

∠C + ∠D = 180°
∠C = 4 × ∠D

Theorems

Linear Pair Theorem
Supplementary Angles Theorem

Suitable Grade Level

Grades 8-10