Chapter 14 — Angle Chasing
Topics in this chapter:
14.1 What Is Angle Chasing?
Angle chasing is the art of systematically finding unknown angles by applying basic geometric facts one step at a time. The problems that follow will help you master this technique. You are asked to find the measure of angles, and you have many tools at your disposal with which to attack these problems.
Strategy. Whenever you are stuck on a problem, come back to the toolkit below and check whether any entry applies to the problem at hand. Often the key insight is simply recognizing which tool to use.
14.2 Your Toolkit
Angle-Chasing Checklist
| # | Tool | # | Tool |
|---|---|---|---|
| 1 | Sum of angles in a triangle ($180°$) | 9 | Isosceles triangles (base angles equal) |
| 2 | Sum of angles in a quadrilateral ($360°$) | 10 | Equilateral triangles (all $60°$) |
| 3 | Inscribed angles intercepting arcs | 11 | Vertical angles (equal) |
| 4 | Angles which together form a straight line ($180°$) | 12 | Similar triangles (equal angles) |
| 5 | Angles around a point ($360°$) | 13 | Congruent triangles (equal angles) |
| 6 | Angles in a right triangle | 14 | Parallel lines and transversal angles |
| 7 | Inscribed angles subtending the same arc | 15 | Exterior angle of a triangle |
| 8 | Angle bisectors | 16 | Perpendicular lines ($90°$) |
How to use the checklist. When you encounter a geometry problem:
- Label every angle you know.
- Scan the checklist and ask: “Does this tool give me a new angle?”
- Write the equation, solve for the unknown angle (or express in terms of others).
- Repeat until you reach the target angle.
Angle chasing is often not deep — it is systematic. The difficulty lies in choosing the right sequence of tools and noticing the relevant configuration.
Common pitfalls:
- Forgetting that an inscribed angle is half the intercepted arc.
- Confusing alternate interior angles with co-interior (same-side) angles at a transversal.
- Not recognizing hidden isosceles triangles formed by radii.
- Assuming angles are equal without proof (avoid “it looks like $45°$”).
Problems to Solve
236. $ABC$ is an isosceles triangle with $AC = BC$. $CBD$ is an isosceles triangle with $CB = DB$. $BD$ meets $AC$ at a right angle. If $\angle A = 57°$, find $\angle D$. (MATHCOUNTS 1986)
237. In $\triangle ABC$, $D$ is some interior point, and $x, y, z, w$ are the measures of angles (in degrees) as shown. Solve for $x$ in terms of $y$, $z$, and $w$. (AHSME 1987)
238. In quadrilateral $ABCD$, $\angle ABC = 110°$, $\angle BCD = 100°$, and angles $\angle BAD$ and $\angle CDA$ are trisected as shown. What is the degree measure of $\angle AFD$? (MATHCOUNTS 1991)
239. In the adjoining figure, $ABCD$ is a square, $ABE$ is an equilateral triangle, and point $E$ is outside square $ABCD$. What is the measure of $\angle AED$? (AHSME 1979)
240. In the figure, triangles $RTS$ and $UTV$ are congruent, $\angle R = 36°$, and $\angle T = 42°$. Find $\angle RQV$. (MATHCOUNTS 1989)
241. In $\triangle ACD$, $\angle A = 50°$ and $\angle CFD = 110°$. If $CE$ bisects $\angle ACD$ and $DB$ is the altitude to $AC$, find $\angle CDF$. (MAΘ 1987)
242. Triangle $ABC$ is isosceles with base $AC$. Points $P$ and $Q$ are respectively on $CB$ and $AB$ such that $AC = AP = PQ = QB$. Find $\angle B$. (AHSME 1961)
243. In a circle with center $O$, $AD$ is a diameter, $ABC$ is a chord, $BO = 5$, and $\angle ABO = CD = 60°$. Find $BC$. (AHSME 1985)
244. In the drawing, $EBC$ is an equilateral triangle and $ABCD$ is a square. Find the measure of $\angle BED$. (MAΘ 1987)
245. In a general triangle $ADE$ (as shown), lines $EB$ and $EC$ are drawn. Show that $x + y + n = a + b + m$. (AHSME 1958)
246. Prove that if the midpoints of the sides of a quadrilateral are vertices of a rectangle, then this quadrilateral is orthodiagonal. (M&IQ 1991)
247. Triangle $PAB$ is formed by $PR$, $PT$, and $AB$, all tangent to circle $O$. If $\angle APB = 40°$, find $\angle AOB$. (AHSME 1956)
248. In the figure, $O$ is the center of the circle, $\angle EAD = 40°$, and $\overset{\frown}{ED} = 40°$. Find $\angle DAB$. (MAΘ 1987)
249. Given triangle $PQR$ with $RS$ bisecting $\angle R$, $PQ$ extended to $D$, and $CD \perp RS$. Show that $m = (p + q)/2$. (AHSME 1954)
250. Quadrilaterals $ABCG$ and $FGDE$ are parallelograms. Points $A, B, C, D, E, F$ are points on the circle. Determine $AB + ED$. (MAΘ 1990)
251. In the figure, $PA$ is tangent to semicircle $SAR$; $PB$ is tangent to semicircle $RBT$; $SRT$ is a straight line; the arc measures (not lengths!) are indicated in the figure. Show that $\angle APB = c + d$. (AHSME 1955)