Math Problem Statement

ABCD is a parallelogram. Find the measure of angle B using algebra.

ABCD is a parallelogram. Find the measure of angle B using algebra.

Solution

To solve the problem involving a parallelogram ABCDABCD and find the measure of angle BB, we first analyze the given details in the image:

Observations:

  1. ABCDABCD is a parallelogram, so:

    • Opposite sides are equal (AB=CDAB = CD and BC=ADBC = AD).
    • Opposite angles are equal (A=C\angle A = \angle C and B=D\angle B = \angle D).
    • Consecutive angles are supplementary (A+B=180\angle A + \angle B = 180^\circ).
  2. The expressions on the sides:

    • AB=2x+40AB = 2x + 40
    • BC=5yBC = 5y
    • CD=4x+10CD = 4x + 10
    • AD=2x+5AD = 2x + 5

Steps to Solve:

  1. Equating Opposite Sides:

    • Since AB=CDAB = CD, we have: 2x+40=4x+102x + 40 = 4x + 10 Simplify: 4010=4x2x40 - 10 = 4x - 2x 30=2x    x=1530 = 2x \implies x = 15

    • Since BC=ADBC = AD, we have: 5y=2x+55y = 2x + 5 Substitute x=15x = 15: 5y=2(15)+55y = 2(15) + 5 5y=30+5    5y=35    y=75y = 30 + 5 \implies 5y = 35 \implies y = 7

  2. Determine Angles:

    • In parallelogram ABCDABCD, consecutive angles A+B=180\angle A + \angle B = 180^\circ. Using the parallelogram property and side information, determine expressions or further relationships for angles if explicitly given (or angles derived geometrically, e.g., supplementary).

Let me confirm if more details are required for angle calculation—please specify how the angle relationship ties to the problem. Would you like me to verify supplementary reasoning or other properties?


Would you like further clarifications?

Relative Questions:

  1. What are the general properties of a parallelogram regarding its angles and diagonals?
  2. Why is the relationship AB=CDAB = CD important in solving for xx?
  3. How can you use supplementary angles to find individual angle measures in a parallelogram?
  4. Can you verify x=15x = 15 and y=7y = 7 using alternative methods?
  5. How do angle properties help solve problems involving parallelograms in geometry?

Tip: Always verify your solution by substituting the values back into the original expressions to ensure consistency.

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometric Properties of Parallelograms
Supplementary Angles
Equations

Formulas

AB = CD
BC = AD
Angle A + Angle B = 180°

Theorems

Opposite sides of a parallelogram are equal
Opposite angles of a parallelogram are equal
Consecutive angles of a parallelogram are supplementary

Suitable Grade Level

Grades 7-9