Examkin

Question

Difficulty: EasyTriangle Properties and Angle Theorems

In △PQR\triangle PQR, the lengths of sides PQPQ and PRPR are equal. If the measure of ∠P\angle P is 70∘70^\circ, what is the measure, in degrees, of ∠Q\angle Q?

Answer: 55 degrees

Answer

The measure of ∠Q\angle Q is 55∘55^\circ.
Since PQ=PRPQ = PR, △PQR\triangle PQR is an isosceles triangle with base QRQR, which means the base angles ∠Q\angle Q and ∠R\angle R have equal measures. The sum of the interior angles of a triangle is 180∘180^\circ. Setting up the equation gives 70∘+2(∠Q)=180∘70^\circ + 2(\angle Q) = 180^\circ. Subtracting 70∘70^\circ from both sides yields 2(∠Q)=110∘2(\angle Q) = 110^\circ. Dividing by 2, we find that the measure of ∠Q\angle Q is 55∘55^\circ.

Step-by-Step Solution

1
Identify the properties of the given triangle.
△PQR\triangle PQR is an isosceles triangle with base QRQR and base angles ∠Q=∠R\angle Q = \angle R.
A triangle with two equal sides is isosceles, and the angles opposite those sides are equal in measure.
2
Apply the triangle angle sum theorem.
∠P+∠Q+∠R=180∘\angle P + \angle Q + \angle R = 180^\circ
The sum of the measures of the interior angles of any triangle is always 180∘180^\circ.
3
Substitute the known values and solve for ∠Q\angle Q.
70∘+2(∠Q)=180∘  ⟹  2(∠Q)=110∘  ⟹  ∠Q=55∘70^\circ + 2(\angle Q) = 180^\circ \implies 2(\angle Q) = 110^\circ \implies \angle Q = 55^\circ
Substituting ∠P=70∘\angle P = 70^\circ and ∠R=∠Q\angle R = \angle Q allows us to solve the linear equation for the unknown angle measure.

Key Concept

Isosceles Triangle Properties and Triangle Angle Sum Theorem
Estimated Time:45s
Rate this question

Topics

Triangle Properties and Angle Theorems

More questions from this topic