Geometry · Chapter 29

Angle Chasing

Straight lines, triangles, parallels, polygons and circles: five simple angle rules that, chained together, crack surprisingly hard contest problems.

What's this chapter about?

29.1a Lines, Triangles and Isosceles Chains
29.1b Polygon Angles
29.1c Angles in Circles
29.2 Practice Problems

You'll need

Nothing beyond solid middle-school math. This is where it all starts.

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29.1a0%

Lines, Triangles and Isosceles Chains

In this section you will be able to chase angles with straight lines, crossing lines, parallels, triangle sums and isosceles triangles.

A detective never sees the crime; she follows clues. Five rules (straight lines, triangles, parallels, polygons and circles) are all the clues you get.

The book’s three ground rules come first. Angles on a straight line add to $180^\circ$ . Angles in a triangle add to $180^\circ$ . And in a triangle, equal sides face equal angles (and equal angles face equal sides): that is an isosceles triangle.

Angle chasing means naming one unknown angle, say $x$ , and writing every other angle in terms of it. Look: an exterior angle of a triangle equals the sum of the two far angles. That one fact powers most chains.

Exploreplay first, formulas later

Try this: Drag any corner of the triangle. The torn-off corners always line up along a straight line. So what do the three angles add to?
Try this: Switch to Parallel lines and tilt the transversal. Which angles always match, and which pairs add to 180°?
Try this: Open Isosceles chain and label the angles x, 2x, 3x from the top. Press Close the chain: why must 3x equal (180° − x) ÷ 2?

Name it

Concept 29.1.1 · Complementary Angles

Complementary angles are a pair of angles with a sum of $90^\circ$ .

Concept 29.1.2 · Supplementary Angles

Supplementary angles are a pair of angles with a sum of $180^\circ$ .

Concept 29.1.3 · Intersecting Lines

When two lines intersect, the vertical angles are equal. Vertical angles are each of the pairs of opposite angles made by two intersecting lines. "Vertical" here means they share the same vertex (corner point), not the usual meaning of up-down.

Concept 29.1.4 · Parallel Lines

When a line crosses two parallel lines, corresponding angles are equal. So every angle at the crossings is either $a$ or $b$ , with $a + b = 180^\circ$ .

Watch itpredict, then reveal

Example 29.1

AMC 10

Triangle $ABC$ has $AB = 2 \cdot AC$ . Let $D$ and $E$ be on $\overline{AB}$ and $\overline{BC}$ , respectively, such that $\angle BAE = \angle ACD$ . Let $F$ be the intersection of segments $AE$ and $CD$ , and suppose that $\triangle CFE$ is equilateral. What is $\angle ACB$ ?

1
Call the two equal angles $x$ : $\angle BAE = \angle ACD = x$ . Triangle $CFE$ is equilateral. What is $\angle FCE$ ?

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Example 29.2

AIME

In $\triangle ABC$ with $AB = AC$ , point $D$ lies strictly between $A$ and $C$ on side $\overline{AC}$ , and point $E$ lies strictly between $A$ and $B$ on side $\overline{AB}$ such that $AE = ED = DB = BC$ . The degree measure of $\angle ABC$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

1
Call the apex angle $\angle A = x$ . Triangle $AED$ has $AE = ED$ . Which angle equals $x$ , and what is $\angle ADE$ ?

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Problem 29.2.15(additional)

AMC 10

The sum of two angles of a triangle is $\frac{6}{5}$ of a right angle, and one of these two angles is $30^\circ$ larger than the other. What is the degree measure of the largest angle in the triangle?

Problem 29.2.10(additional)

AMC 10

This problem also appears in the book as 15.1.9.

How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?

Contest cornerreal contest problems

Problem 29.2.13(additional)

AHSME

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 $\measuredangle AED$ in degrees?

Problem 29.2.12(additional)

AMC 10/12

As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$ . Point $F$ lies on $\overline{AD}$ so that $DE = DF$ , and $ABCD$ is a square. What is the degree measure of $\angle AFE$ ?

Problem 29.2.8(additional)

AHSME

$\triangle ABC$ is isosceles with base $AC$ . Points $P$ and $Q$ are respectively in $CB$ and $AB$ and such that $AC = AP = PQ = QB$ . The number of degrees in $\angle B$ is:

Problem 29.2.19(additional)

AMC 10Challenge

Quadrilateral $ABCD$ has $AB = BC = CD$ , angle $ABC = 70^\circ$ and angle $BCD = 170^\circ$ . What is the measure of angle $BAD$ ?

Checkpoint for section 29.1a

Get 2 of 3 right to master this section. One try each, no hints.

29.1b0%

Polygon Angles

In this section you will be able to find interior and exterior angles of any polygon, and spot the ones worth memorising.

Walk once around any polygon and you turn exactly one full circle. That single fact gives you every exterior angle, and every interior angle with it.

Draw all the diagonals from one corner of an $n$ -gon. You get $n - 2$ triangles, so the interior angles add to $(n - 2) \cdot 180^\circ$ . In a regular polygon all $n$ corners share that sum equally.

Look: the exterior angle is often the faster route. It is just $360^\circ \div n$ , and interior $= 180^\circ -$ exterior.

Exploreplay first, formulas later

Try this: Step the number of sides from 3 up to 10. How many triangles does one corner make each time? Why does that give (n − 2) · 180°?
Try this: Turn on the Table of angles. Check each row: does interior + exterior always make 180°?
Try this: Press Walk the perimeter. How far have you turned when you get back home?

Name it

Theorem 29.1.5

For a polygon with $n$ sides:

\text{Sum of interior angles} = (n-2)\cdot 180^\circ

\text{Interior angle of a regular polygon} = \frac{(n-2)}{n}\cdot 180^\circ

\text{Exterior angle of a regular polygon} = \frac{360^\circ}{n}

Fact 29.1.6 · Important Interior Angles

Important interior angles of regular polygons:

Number of sides	Interior angle
3	$60^\circ$
4	$90^\circ$
5	$108^\circ$
6	$120^\circ$
8	$135^\circ$
9	$140^\circ$
10	$144^\circ$

Try it

Problem 29.2.14(additional)

AMC 10

The angles of quadrilateral $ABCD$ satisfy $\angle A = 2\angle B = 3\angle C = 4\angle D$ . What is the degree measure of $\angle A$ , rounded to the nearest whole number?

Problem 29.2.16(additional)

AMC 10

This problem also appears in the book as 15.1.8.

Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?

Contest cornerreal contest problems

Problem 29.2.4

AMC 10

Square $ABCD$ is rotated $20^\circ$ clockwise about its center to obtain square $EFGH$ , as shown below. What is the degree measure of $\angle EAB$ ?

Problem 29.2.5

MATHCOUNTS

A square is located in the interior of a regular hexagon, and certain vertices are labeled as shown. What is the degree measure of $\angle ABC$ ?

Problem 29.2.7

AMC 10

The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $x$ ?

Checkpoint for section 29.1b

Get 2 of 3 right to master this section. One try each, no hints.

29.1c0%

Angles in Circles

In this section you will be able to use inscribed angles, chords, tangents and cyclic quadrilaterals to chase angles in circles.

Stand anywhere on the far side of a circle and look at a fixed arc. You see it at the same angle from every spot: exactly half of the angle at the center.

Circles hand you equal angles for free. Every angle in this section comes from one idea: an angle is half the arc it looks at. Chords crossing inside average two arcs; secants meeting outside take half their difference; a tangent and a chord take half the arc between them.

Look: radii are all equal, so drawing them creates isosceles triangles. And a radius meets a tangent at $90^\circ$ .

Exploreplay first, formulas later

Try this: Drag P anywhere on the big arc. Does the angle at P change? Compare it with the central angle at O.
Try this: Drag A and B until AB is a diameter. What angle does P see now?
Try this: Drag P onto the small arc between A and B. What happens to the angle, and which arc does it look at now?

Name it

Theorem 29.1.7 · Inscribed Arc Theorem

The angle an arc makes at the center (the arc's angle) is double the angle it makes at any point on the edge of the circle.

\text{central angle} = 2 \times \text{inscribed angle}

Corollary 29.1.8 · Inscribed Right Triangle

An inscribed triangle with a diameter as one side is always a right triangle.

Definition 29.1.9 · Chord

A chord is a line segment between any two distinct points on the circle. The diameter of the circle is the longest chord in the circle.

Theorem 29.1.10

The perpendicular bisector of any chord passes through the center. In the figure, the perpendicular bisectors of $AB$ and $CD$ intersect at the center $O$ .

Corollary 29.1.11

Congruent chords are equidistant from the center of a circle.
If two chords in a circle are congruent, then their intercepted arcs are congruent.
If two chords in a circle are congruent, then they determine two central angles that are congruent.

Theorem 29.1.12

Two secants from an outside point $P$ cut the circle at $A, B$ and $C, D$ . The angle at $P$ is half the difference of the two arcs it cuts off:

\angle APC = \frac{\overset{\frown}{BD} - \overset{\frown}{AC}}{2}

Theorem 29.1.13

If two chords $AB$ and $CD$ intersect at $P$ , then $\angle BPC$ and $\angle APD$ are equal to the average of the two arcs:

\angle BPC = \angle APD = \frac{\overset{\frown}{BC} + \overset{\frown}{AD}}{2}

Theorem 29.1.14

If a tangent line through $R$ touches the circle at $Q$ , and a chord $QP$ is drawn, then $\angle RQP$ is equal to half the arc $QP$ inside the angle.

Remark 29.1.15

Circles are really useful for angle chasing, so keep an eye out for the inscribed arc theorem. It can be used in many angle chasing problems.

Remark 29.1.16

A useful trick for angle chasing problems with regular polygons: draw a circle around the polygon and use the inscribed arc theorem.

Theorem 29.1.17

Equal chords mark out equal arcs. This means that if you have 2 chords of the same length, the arcs (and sectors) of the circle they mark out are equal.

Definition 29.1.18 · Tangent

A tangent is any line from a point external to the circle that just touches the circle.

Theorem 29.1.19 · Right Angle Tangency Point

If you connect the center of a circle to the point where the circle and a line are tangent, they form a right angle.

Theorem 29.1.21 · Properties of a Cyclic Quadrilateral

In a cyclic quadrilateral (all four corners on one circle), the sum of opposite angles is $180^\circ$ : $p + q = 180^\circ$ and $r + s = 180^\circ$ .

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Example 29.3

AMC 10

Let $\triangle ABC$ be an isosceles triangle with $BC = AC$ and $\angle ACB = 40^\circ$ . Construct the circle with diameter $\overline{BC}$ , and let $D$ and $E$ be the other intersection points of the circle with the sides $\overline{AC}$ and $\overline{AB}$ , respectively. Let $F$ be the intersection of the diagonals of the quadrilateral $BCDE$ . What is the degree measure of $\angle BFC$ ?

1
$BC$ is a diameter and $D$ is on the circle. What is $\angle BDC$ ? (And $\angle BEC$ ?)

Predict first:
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Contest cornerreal contest problems

Problem 29.2.2

AMC

Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$ . The circles intersect at two points, one of which is $E$ . What is the degree measure of $\angle CED$ ?

Problem 29.2.6

AMC 10

In the given circle, the diameter $\overline{EB}$ is parallel to $\overline{DC}$ , and $\overline{AB}$ is parallel to $\overline{ED}$ . The angles $AEB$ and $ABE$ are in the ratio $4 : 5$ . What is the degree measure of angle $BCD$ ?

Problem 29.2.3

AMC 10

Abdul and Chiang are standing 48 feet apart in a field. Bharat is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures $60^\circ$ . What is the square of the distance (in feet) between Abdul and Bharat?

Problem 29.2.17(additional)

AHSME

An acute isosceles triangle, $ABC$ , is inscribed in a circle. Through $B$ and $C$ , tangents to the circle are drawn, meeting at point $D$ . If $\angle ABC = \angle ACB = 2\angle D$ and $x$ is the radian measure of $\angle A$ , then $x =$

Checkpoint for section 29.1c

Get 2 of 3 right to master this section. One try each, no hints.

29.2

Practice Problems

In this section you will be able to mix every angle-chasing tool from this chapter on harder contest problems.

These problems need several clues at once: polygons, isosceles chains, symmetry and even light beams. Every problem has three hints if you get stuck.

Want every problem from this chapter on one page? Open the problem set.

Contest cornerreal contest problems

Problem 29.2.1

AMC 10

Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\sqrt{3}$ , and the lines $BC, DE,$ and $FA$ determine a triangle with area $324\sqrt{3}$ . The perimeter of hexagon $ABCDEF$ can be expressed as $m + n\sqrt{p}$ , where $m, n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m + n + p$ ?

Problem 29.2.9(additional)

AHSME

In this diagram $AB$ and $AC$ are the equal sides of an isosceles $\triangle ABC$ , in which is inscribed equilateral $\triangle DEF$ . Designate $\angle BFD$ by $a$ , $\angle ADE$ by $b$ , and $\angle FEC$ by $c$ . Then:

Problem 29.2.11(additional)

MATHCOUNTS

Concave quadrilateral $ABCD$ is symmetric about the line $AC$ . The measures of angles $DAB$ and $ABC$ are $84$ degrees and $32$ degrees, respectively. The dashed line segments bisect angles $ABC$ and $ADC$ . What is the degree measure of the acute angle at which the two dashed line segments intersect?

Problem 29.2.18(additional)

AIMEChallenge

Triangle $ABC$ is isosceles with $AC = BC$ and $\angle ACB = 106^\circ$ . Point $M$ is in the interior of the triangle so that $\angle MAC = 7^\circ$ and $\angle MCA = 23^\circ$ . Find the number of degrees in $\angle CMB$ .

Problem 29.2.20(additional)

AIMEChallenge

A beam of light strikes $\overline{BC}$ at point $C$ with angle of incidence $\alpha = 19.94^\circ$ and reflects with an equal angle of reflection as shown. The light beam continues its path, reflecting off line segments $\overline{AB}$ and $\overline{BC}$ according to the rule: angle of incidence equals angle of reflection. Given that $\beta = \alpha/10 = 1.994^\circ$ and $AB = BC$ , determine the number of times the light beam will bounce off the two line segments. Include the first reflection at $C$ in your count.

Formula cards

Concept 29.1.1 · Complementary Angles

Complementary angles are a pair of angles with a sum of $90^\circ$ .

Concept 29.1.2 · Supplementary Angles

Supplementary angles are a pair of angles with a sum of $180^\circ$ .

Concept 29.1.3 · Intersecting Lines

Concept 29.1.4 · Parallel Lines

When a line crosses two parallel lines, corresponding angles are equal. So every angle at the crossings is either $a$ or $b$ , with $a + b = 180^\circ$ .

Theorem 29.1.5

For a polygon with $n$ sides:

\text{Sum of interior angles} = (n-2)\cdot 180^\circ

\text{Interior angle of a regular polygon} = \frac{(n-2)}{n}\cdot 180^\circ

\text{Exterior angle of a regular polygon} = \frac{360^\circ}{n}

Fact 29.1.6 · Important Interior Angles

Important interior angles of regular polygons:

Number of sides	Interior angle
3	$60^\circ$
4	$90^\circ$
5	$108^\circ$
6	$120^\circ$
8	$135^\circ$
9	$140^\circ$
10	$144^\circ$

Theorem 29.1.7 · Inscribed Arc Theorem

The angle an arc makes at the center (the arc's angle) is double the angle it makes at any point on the edge of the circle.

\text{central angle} = 2 \times \text{inscribed angle}

Corollary 29.1.8 · Inscribed Right Triangle

An inscribed triangle with a diameter as one side is always a right triangle.

Theorem 29.1.10

The perpendicular bisector of any chord passes through the center. In the figure, the perpendicular bisectors of $AB$ and $CD$ intersect at the center $O$ .

Corollary 29.1.11

Congruent chords are equidistant from the center of a circle.
If two chords in a circle are congruent, then their intercepted arcs are congruent.
If two chords in a circle are congruent, then they determine two central angles that are congruent.

Theorem 29.1.12

Two secants from an outside point $P$ cut the circle at $A, B$ and $C, D$ . The angle at $P$ is half the difference of the two arcs it cuts off:

\angle APC = \frac{\overset{\frown}{BD} - \overset{\frown}{AC}}{2}

Theorem 29.1.13

If two chords $AB$ and $CD$ intersect at $P$ , then $\angle BPC$ and $\angle APD$ are equal to the average of the two arcs:

\angle BPC = \angle APD = \frac{\overset{\frown}{BC} + \overset{\frown}{AD}}{2}

Theorem 29.1.14

If a tangent line through $R$ touches the circle at $Q$ , and a chord $QP$ is drawn, then $\angle RQP$ is equal to half the arc $QP$ inside the angle.

Theorem 29.1.17

Equal chords mark out equal arcs. This means that if you have 2 chords of the same length, the arcs (and sectors) of the circle they mark out are equal.

Theorem 29.1.19 · Right Angle Tangency Point

If you connect the center of a circle to the point where the circle and a line are tangent, they form a right angle.

Theorem 29.1.21 · Properties of a Cyclic Quadrilateral

In a cyclic quadrilateral (all four corners on one circle), the sum of opposite angles is $180^\circ$ : $p + q = 180^\circ$ and $r + s = 180^\circ$ .

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