In a ΔPQR, M lies on PR and between P and R such that QR=QM=PM. If ∠MQR=40^∘, then find ∠P. (1)

step 1 Hello. Today we are doing this question in which a figure is given in this triangle PQR M lies o

In a ΔPQR, M lies on PR and between P and R such that...

In a $\Delta \mathrm{PQR}, \mathrm{M}$ lies on $\mathrm{PR}$ and between $\mathrm{P}$ and $\mathrm{R}$ such that $\mathrm{QR}=\mathrm{QM}=\mathrm{PM}$. If $\angle \mathrm{MQR}=40^{\circ}$, then find $\angle P$.
(1) $35^{\circ}$
(2) $25^{\circ}$
(3) $45^{\circ}$
(4) $55^{\circ}$

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Key Concepts

Isosceles Triangle Properties

An isosceles triangle has at least two congruent sides, which implies that the base angles opposite those sides are equal. This principle helps in establishing relationships between different angles in a geometric figure, particularly when equal segments are involved.

Angle Sum Property of a Triangle

In any triangle, the measures of the three interior angles always add up to 180°. This fundamental rule allows one to determine unknown angles when the measures of the other angles are known or can be expressed in terms of one another.

Angle Chasing

Angle chasing involves deducing the measures of unknown angles by logically applying known geometric properties and relationships, such as the equal angles in an isosceles triangle and the fact that the sum of angles in a triangle is 180°. This systematic approach is essential for solving many geometry problems.

Segment Partitioning in Triangles

Introducing a point to split one side of a triangle into segments with specific relations, such as having equal lengths, often leads to the formation of additional isosceles triangles. Recognizing and utilizing these subdivisions is crucial for breaking down complex geometric relationships into simpler, more manageable components.

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