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Zorluk: OrtaTriangle Properties and Angle Theorems

In △PQR\triangle PQR, the measure of ∠P\angle P is 50∘50^\circ, and the measure of the exterior angle at vertex QQ is 110∘110^\circ. If the bisector of ∠PRQ\angle PRQ intersects side PQPQ at point SS, what is the measure, in degrees, of ∠PRS\angle PRS?

Cevap: 30 degrees

Cevap

The measure of ∠PRS\angle PRS is 30∘30^\circ.
By using the linear pair relationship, the interior angle ∠PQR\angle PQR is found to be 70∘70^\circ. Applying the triangle angle sum theorem, the third interior angle ∠PRQ\angle PRQ is 180∘−(50∘+70∘)=60∘180^\circ - (50^\circ + 70^\circ) = 60^\circ. The angle bisector RSRS divides this angle into two equal parts, resulting in a measure of 30∘30^\circ for ∠PRS\angle PRS.

Adım Adım Çözüm

1
Find the interior angle at vertex QQ
∠PQR=70∘\angle PQR = 70^\circ
The interior and exterior angles at a vertex are supplementary, summing to 180∘180^\circ.
2
Find the measure of interior angle ∠PRQ\angle PRQ
∠PRQ=60∘\angle PRQ = 60^\circ
The sum of the interior angles in any triangle is 180∘180^\circ.
3
Calculate the measure of the bisected angle ∠PRS\angle PRS
∠PRS=30∘\angle PRS = 30^\circ
An angle bisector divides the angle into two equal measures.

Anahtar Kavram

Triangle Angle Sum Theorem and Exterior Angle Relationships
Tahmini Süre:1m 30s
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Konular

Triangle Properties and Angle Theorems

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