Geometry › Lines and angles

Angle chasing

Solve the first problem 16 problems

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1. Centre of square on hypotenuse ●●○○○ 20 ptsLet $ABC$ be a right triangle with the right angle at $B$. Let $ACDE$ be a square drawn outside triangle $ABC$ on side $AC$. If $M$ is the centre of this square, find $\angle MBC$ in degrees.
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 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.
5. Angle NCB in a triangle ●●●○○ 50 ptsIn $\triangle ABC$, $\angle ABC=50^{\circ}$ and $\angle ACB=20^{\circ}$. Point $N$ is inside $\triangle ABC$ with $\angle NAB=40^{\circ}$ and $\angle NBC=30^{\circ}$. Find the measure of $\angle NCB$ in degrees.
6. Angle PCQ in a unit square ●●●○○ 50 ptsGiven a square $ABCD$ with side length $1$, points $P$ and $Q$ lie on sides $AB$ and $AD$, respectively. If the perimeter of $\triangle APQ$ is $2$, find the angle $\angle PCQ$ in degrees.
7. 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.

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