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8 problems match “isosceles triangles”. Clear

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1. Altitude of an isosceles triangle ●○○○○ 10 ptsIn triangle $ABC$ with $AB=AC=25$ and $BC=14$, let $D$ be the foot of the altitude from $A$ to $BC$. Find the length of $AD$.
2. Incircle touch points angle ●●○○○ 20 ptsIn triangle $ABC$, $\angle BAC = 72^\circ$. The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D$, $E$, and $F$, respectively. Find $\angle EDF$, in degrees.
3. Angle in two isosceles triangles ●●○○○ 20 ptsTriangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$. Point $D$ is inside $\triangle ABC$, $\angle ABC = 40^\circ$, and $\angle ADC = 140^\circ$. Find the measure of $\angle BAD$ in degrees.
4. Angle in a regular pentagon ●●○○○ 20 ptsIn a regular pentagon $PQRST$, find the measure of $\angle PRS$ in degrees.
5. Angle in pentagon with inner triangle ●●○○○ 20 ptsIn a regular pentagon $ABCDE$, an equilateral triangle $ABM$ is drawn so that $M$ lies inside the pentagon. Determine the size of the angle $BCM$ in degrees.
6. Angle ACB with AB equal to CD ●●●○○ 50 ptsIn triangle $ABC$, point $D$ lies on $BC$ such that $\angle BAD = 60^\circ$, $\angle B = 20^\circ$ and $AB = CD$. Find $\angle ACB$ in degrees.
7. Segment LM in a 30-60-90 triangle ●●●○○ 50 ptsIn triangle $ABC$, $\angle B=30^{\circ}$ and $\angle A=90^{\circ}$. On side $AC$, point $K$ is marked, and on side $BC$, points $L$ and $M$ are marked such that $KL=KM$ (point $L$ lies on segment $BM$). Given $AK=4$, $BL=31$, $MC=3$, find the length of segment $LM$.
8. Angle APC from given cevian angles ●●●●○ 100 pts$P$ is a point inside triangle $ABC$ such that $\angle PBC=30^\circ$, $\angle PBA=8^\circ$ and $\angle PAB=\angle PAC=22^\circ$. Find $\angle APC$, in degrees.