Math Problem Statement

In △PQR, QR ≅ PQ and m∠Q = 159°. Find m∠P.

Solution

To solve this problem, we analyze the given information about triangle PQR\triangle PQR:

Given:

  1. QRPQQR \cong PQ (so the triangle is isosceles).
  2. mQ=159m\angle Q = 159^\circ.
  3. We need to find mPm\angle P (labeled xx^\circ).

Step-by-step Solution:

  1. Sum of angles in a triangle: The sum of the internal angles in any triangle is always 180180^\circ. Therefore: mP+mQ+mR=180.m\angle P + m\angle Q + m\angle R = 180^\circ.

  2. Isosceles Triangle Property: Since QRPQQR \cong PQ, the angles opposite these equal sides are also equal. Thus: mP=mR=x.m\angle P = m\angle R = x^\circ.

  3. Substitute known values: Replace mQ=159m\angle Q = 159^\circ and simplify: x+159+x=180.x + 159^\circ + x = 180^\circ.

    Combine like terms: 2x+159=180.2x + 159^\circ = 180^\circ.

  4. Solve for xx: Subtract 159159^\circ from both sides: 2x=180159.2x = 180^\circ - 159^\circ. 2x=21.2x = 21^\circ.

    Divide by 2: x=212=10.5.x = \frac{21^\circ}{2} = 10.5^\circ.


Final Answer:

The measure of P\angle P is: 10.5\boxed{10.5^\circ}

This corresponds to option B.


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Related Questions:

  1. What are the properties of isosceles triangles?
  2. How do you calculate the angles of a triangle given one angle?
  3. What is the sum of the interior angles of a polygon?
  4. How do you solve for missing angles in triangles with congruent sides?
  5. Can you explain exterior angles of triangles and their relationships?

Tip: Always use triangle symmetry to simplify calculations in isosceles and equilateral triangles!

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Math Problem Analysis

Mathematical Concepts

Triangle Angle Sum
Isosceles Triangle Properties
Basic Algebra

Formulas

Sum of angles in a triangle: ∠P + ∠Q + ∠R = 180°

Theorems

Isosceles Triangle Theorem

Suitable Grade Level

Grades 8-10