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 :
Given:
- — This implies is an isosceles triangle with two sides equal.
- .
- We need to find .
Step-by-step Solution:
-
Properties of an Isosceles Triangle: In an isosceles triangle, the angles opposite the equal sides are also equal. Since , the angles and are equal.
Therefore:
-
Sum of Angles in a Triangle: The sum of all angles in a triangle is always . For :
Substituting and , we get:
-
Simplify the Equation: Combine like terms:
-
Solve for : Subtract from both sides:
Divide by 2:
Final Answer:
Would you like further details or additional explanations? Here are 5 related questions to expand on this concept:
- How do you determine if a triangle is isosceles using given side or angle information?
- What are some real-world applications of isosceles triangle properties?
- How can you prove the base angles of an isosceles triangle are equal geometrically?
- If were different, how would the calculations change?
- 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