Math Problem Statement

In triangle, P, Q, R, comma△PQR, start overline, Q, R, end overline, \cong, start overline, P, Q, end overlineQR​≅PQ​ and m, angle, Q, equals, 84, degrees, .m∠Q=84∘. Find m, angle, P, .m∠P.

Solution

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

Given:

  1. QRPQ\overline{QR} \cong \overline{PQ} — This implies PQR\triangle PQR is an isosceles triangle with two sides equal.
  2. mQ=84m\angle Q = 84^\circ.
  3. We need to find mPm\angle P.

Step-by-step Solution:

  1. Properties of an Isosceles Triangle: In an isosceles triangle, the angles opposite the equal sides are also equal. Since QRPQ\overline{QR} \cong \overline{PQ}, the angles P\angle P and R\angle R are equal.

    Therefore: mP=mR.m\angle P = m\angle R.

  2. Sum of Angles in a Triangle: The sum of all angles in a triangle is always 180180^\circ. For PQR\triangle PQR: mP+mQ+mR=180.m\angle P + m\angle Q + m\angle R = 180^\circ.

    Substituting mQ=84m\angle Q = 84^\circ and mP=mRm\angle P = m\angle R, we get: mP+84+mP=180.m\angle P + 84^\circ + m\angle P = 180^\circ.

  3. Simplify the Equation: Combine like terms: 2mP+84=180.2m\angle P + 84^\circ = 180^\circ.

  4. Solve for mPm\angle P: Subtract 8484^\circ from both sides: 2mP=96.2m\angle P = 96^\circ.

    Divide by 2: mP=48.m\angle P = 48^\circ.


Final Answer:

mP=48.m\angle P = 48^\circ.

Would you like further details or additional explanations? Here are 5 related questions to expand on this concept:

  1. How do you determine if a triangle is isosceles using given side or angle information?
  2. What are some real-world applications of isosceles triangle properties?
  3. How can you prove the base angles of an isosceles triangle are equal geometrically?
  4. If mQm\angle Q were different, how would the calculations change?
  5. What is the relationship between the side lengths and angles in an equilateral triangle?

Tip: In any isosceles triangle, always look for equal angles opposite the congruent sides—it simplifies many problems!

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Properties
Isosceles Triangle

Formulas

Sum of angles in a triangle: m∠P + m∠Q + m∠R = 180°
Isosceles triangle property: Angles opposite equal sides are equal

Theorems

Sum of Angles in a Triangle Theorem
Base Angles Theorem of Isosceles Triangle

Suitable Grade Level

Grades 7-10