Geometry · Chapter 23

Angle Chasing

Hunt down a missing angle one clue at a time: straight lines, triangles, parallel lines, polygons and circles all leave footprints.

What's this chapter about?

23.1 Angle Chasing Basics
23.2 Inscribed Angles
23.3 Polygons
23.4 Advanced Circle Angle Chasing Theorems (Optional)
23.5 Practice Problems

You'll need

Nothing! This is where it all starts. If you can multiply, you're ready.

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Angle Chasing Basics

In this section you will be able to find a missing angle by chaining simple facts: triangles add to 180°, straight lines add to 180°, and some angles are always equal.

A detective never sees the crime. She follows clues, one at a time, until the answer is cornered. Angle chasing works the same way: each angle you find is a clue to the next.

You only need a few tools. A triangle’s angles add to 180°. Angles on a straight line add to 180°. Opposite angles at a crossing are equal, and so are matching angles on parallel lines.

Look:Whenever two sides of a triangle are equal, two angles are equal too. Hunt for equal sides: radii, square sides, equilateral sides.

Exploreplay first, formulas later

Try this: Drag the corners of the triangle anywhere. Can you make the three angles add to anything other than 180°? Turn on Tear off the corners to see why.
Try this: Press Isosceles and drag A up and down. What stays true about the two bottom angles?
Try this: Press Crossing lines. Which angles are always equal? What do two neighbours add up to?
Try this: Press Parallel lines and tilt the transversal. How many different angle sizes do you ever see?

Name it

Concept 23.1.1 · Angle Chasing Tricks

Sum of Angles in Triangle is $180^\circ$
A triangle with 2 angles equal will have their corresponding sides equal and a triangle with 2 sides equal will have their corresponding angles equal (isosceles triangle)
Opposite angles in intersecting lines are equal
Corresponding angles in parallel lines are equal
The angle made by the arc at the center of the circle is double the angle made by the arc at the boundary of the circle

Concept 23.1.2 · Complementary Angle

Complementary angles are a pair of angles with the sum of 90 degrees:

a + b = 90^\circ

Concept 23.1.3 · Supplementary Angle

Supplementary angles are a pair of angles with the sum of 180 degrees:

a + b = 180^\circ

Concept 23.1.4 · 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" in this case means they share the same Vertex (corner point), not the usual meaning of up-down.

Concept 23.1.5 · Parallel Lines

Corresponding angles equal: when a line crosses two parallel lines, the angles in matching positions at the two crossings are the same.

Watch itpredict, then reveal

Example 23.1

AMC 8

The degree measure of angle $A$ is

1
Find the triangle that has the $40^\circ$ corner and the $100^\circ$ angle. Its third corner is the right end of the horizontal line. How big is that corner?

Predict first:
Your prediction

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Example 23.2

EMCC

In acute $\triangle ABC$ , D and E are points inside triangle ABC such that $DE \parallel BC$ , $B$ is closer to $D$ than it is to $E$ , $\angle AED = 80^\circ$ , $\angle ABD = 10^\circ$ , and $\angle CBD = 40^\circ$ . Find the measure of $\angle BAE$ , in degrees.

1
Extend $AE$ until it hits $BC$ at $F$ . Line $AF$ crosses the parallel lines $DE$ and $BC$ . Which angle at $F$ matches $\angle AED$ ?

Predict first:
Your prediction

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Problem 23.5.1

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 $\angle AED$ in degrees?

Contest cornerreal contest problems

Problem 23.5.3

AMC 8

In $\triangle ABC$ , $D$ is a point on side $\overline{AC}$ such that $BD = DC$ and $\angle BCD$ measures $70^\circ$ . What is the degree measure of $\angle ADB$ ?

Problem 23.5.6

AMC 8

Two angles of an isosceles triangle measure $70^\circ$ and $x^\circ$ . What is the sum of the different possible values of $x$ ?

Problem 23.5.2

AMC 8

In quadrilateral $ABCD$ , sides $\overline{AB}$ and $\overline{BC}$ both have length 10, sides $\overline{CD}$ and $\overline{DA}$ both have length 17, and the measure of angle $ADC$ is $60^\circ$ . What is the length of diagonal $\overline{AC}$ ?

Problem 23.5.10(additional)

EMCC

Point $P$ lies inside triangle $ABC$ such that $\angle PBC = 30^\circ$ and $\angle PAC = 20^\circ$ . If $\angle APB$ is a right angle, find the measure of $\angle BCA$ in degrees.

Problem 23.5.11(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 $\angle AFE$ ?

Checkpoint for section 23.1

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

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Inscribed Angles

In this section you will be able to use the Inscribed Arc Theorem: an angle on the edge of a circle is half the arc it looks at.

Stand anywhere on the edge of a round stage and look at the two ends of the curtain. Surprise: from every spot on the same side, the curtain fills exactly the same angle of your view!

The angle at the center is always twice the angle on the edge. So every angle on the edge that looks at the same arc is equal, and an angle that looks at a diameter is always 90°.

Look:Regular polygon? Draw a circle through its corners. Equal sides cut the circle into equal arcs.

Exploreplay first, formulas later

Try this: Drag P around the top of the circle. Does the angle at P change? Compare it with the central angle at O.
Try this: Now drag A or B to make the arc bigger. What happens to both numbers?
Try this: Drag A and B to opposite ends of a diameter. What is the angle at P now, wherever P goes?

Name it

Theorem 23.2.1 · Inscribed Arc Theorem

The angle formed by an arc in the center or the arc angle is double of the angle formed on the edge.

\angle AOB = 2\,\angle APB

Every angle on the edge that looks at the same arc is equal.

Remark 23.2.3

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

Corollary 23.2.4 · Inscribed Right Triangle

Inscribed triangle with diameter as one side is always a right triangle. The diameter cuts off an arc of $180^\circ$ , so the angle opposite it is $\frac{180^\circ}{2} = 90^\circ$ .

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Example 23.3

In regular octagon ABCDEFGH, Find the measure of $\angle ACF$ .

1
Draw the circle through all 8 corners. The corners cut it into 8 equal arcs. How many degrees is each arc?

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Contest cornerreal contest problems

Problem 23.5.5

AMC 8

The circumference of the circle with center $O$ is divided into 12 equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$ ?

Problem 23.5.7

AMC 8

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 $\angle CED$ ?

Checkpoint for section 23.2

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

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Polygons

In this section you will be able to find the angle sum of any polygon and each angle of a regular polygon.

Bees build hexagons, stop signs are octagons, and soccer balls hide pentagons. How big is each corner? You can work it out with triangles.

Draw every diagonal from one corner. An n-sided polygon splits into n − 2 triangles, and each triangle holds 180°.

If you walk around any polygon, you turn one full circle, 360°. In a regular polygon every turn is the same: 360° ÷ n.

Exploreplay first, formulas later

Try this: Turn on Triangles from one corner. How many triangles does a pentagon split into? A hexagon? An octagon?
Try this: Why does that give (n − 2) × 180° for the sum? Check the Sum box as you change the number of sides.
Try this: Turn on Exterior angles and press Walk the perimeter. How far have you turned when you get back home?

Name it

Theorem 23.3.1

\text{Sum of interior angle of a polygon} = (n-2)\cdot 180

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

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

Fact 23.3.2 · Important Interior Angles

Number of sides in regular polygon	Interior Angle of regular polygon
3	60
4	90
5	108
6	120
8	135
9	140
10	144

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Example 23.4

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 $\angle ABC$ ?

1
The square shares a side with the hexagon, so all their sides are equally long. What kind of triangle is $BCD$ ?

Predict first:

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Contest cornerreal contest problems

Problem 23.5.8

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 $x$ ?

Problem 23.5.12(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?

Checkpoint for section 23.3

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

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Advanced Circle Angle Chasing Theorems (Optional)

In this section you will be able to use chords, secants and tangents to turn arcs into angles (optional extras for hard problems).

A bike wheel touches the road at exactly one point. The spoke pointing down to that point always stands straight up from the road. That right angle is a powerful clue.

This section is optional. Each theorem links an angle to the arcs it looks at. Inside the circle you average the arcs. Outside the circle you take half the difference.

Exploreplay first, formulas later

Try this: In Chord bisectors, drag A, B, C and D. Do the dashed lines ever miss the center?
Try this: Press Crossing chords and then Two secants. When P is inside, do you add the arcs or subtract them? And outside?
Try this: Press Tangent + radius and drag P around. What angle does the radius always make with the tangent?

Name it

Definition 23.4.1 · 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 23.4.2

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 23.4.3

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 23.4.4

The angle marked in the diagram is half of the difference of the 2 red arcs.

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

Theorem 23.4.5

If two chords $AB$ and $CD$ intersect at $P$ , then the $\angle BPC$ 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 23.4.6

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

\angle RQP = \frac{\overset{\frown}{QP}}{2}

Theorem 23.4.7

Equal chords mark out equal arcs.

This basically means that if you have 2 chords of the same length, the sector of the circle they mark out will be equal.

Definition 23.4.8 · Tangent

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

Theorem 23.4.9 · Right Angle Tangency Point

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

Try it

Problem 23.5.4

In the figure, point A is the center of the circle, the measure of angle RAS is $74^\circ$ , and the measure of angle RTB is $28^\circ$ . What is the measure of minor arc BR, in degrees?

Checkpoint for section 23.4

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

23.5

Practice Problems

In this section you will be able to mix every angle-chasing tool on the hardest contest problems.

Time for the toughest cases. Draw the figure, label every angle you know, and keep chasing. 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 23.5.13(additional)

EMCC

In triangle $ABC$ , $\angle BAC$ is a right angle and $\angle ACB$ measures $34^\circ$ . Let $D$ be a point on segment $BC$ for which $AC = CD$ , and let the angle bisector of $\angle CBA$ intersect line $AD$ at $E$ . What is the measure of $\angle BED$ ?

Problem 23.5.14(additional)

EMCC

In square ABCD, point E lies on side BC and point F lies on side CD so that triangle AEF is equilateral and inside the square. Point M is the midpoint of segment EF, and P is the point other than E on AE for which PM = FM. The extension of segment PM meets segment CD at Q. What is the measure of $\angle CQP$ , in degrees?

Formula cards

Concept 23.1.1 · Angle Chasing Tricks

Sum of Angles in Triangle is $180^\circ$
A triangle with 2 angles equal will have their corresponding sides equal and a triangle with 2 sides equal will have their corresponding angles equal (isosceles triangle)
Opposite angles in intersecting lines are equal
Corresponding angles in parallel lines are equal
The angle made by the arc at the center of the circle is double the angle made by the arc at the boundary of the circle

Concept 23.1.2 · Complementary Angle

Complementary angles are a pair of angles with the sum of 90 degrees:

a + b = 90^\circ

Concept 23.1.3 · Supplementary Angle

Supplementary angles are a pair of angles with the sum of 180 degrees:

a + b = 180^\circ

Concept 23.1.4 · Intersecting lines

Concept 23.1.5 · Parallel Lines

Corresponding angles equal: when a line crosses two parallel lines, the angles in matching positions at the two crossings are the same.

Theorem 23.2.1 · Inscribed Arc Theorem

The angle formed by an arc in the center or the arc angle is double of the angle formed on the edge.

\angle AOB = 2\,\angle APB

Every angle on the edge that looks at the same arc is equal.

Theorem 23.2.4 · Inscribed Right Triangle

Inscribed triangle with diameter as one side is always a right triangle. The diameter cuts off an arc of $180^\circ$ , so the angle opposite it is $\frac{180^\circ}{2} = 90^\circ$ .

Theorem 23.3.1

\text{Sum of interior angle of a polygon} = (n-2)\cdot 180

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

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

Theorem 23.3.2 · Important Interior Angles

Number of sides in regular polygon	Interior Angle of regular polygon
3	60
4	90
5	108
6	120
8	135
9	140
10	144

Theorem 23.4.2

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

Theorem 23.4.3

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 23.4.4

The angle marked in the diagram is half of the difference of the 2 red arcs.

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

Theorem 23.4.5

If two chords $AB$ and $CD$ intersect at $P$ , then the $\angle BPC$ 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 23.4.6

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

\angle RQP = \frac{\overset{\frown}{QP}}{2}

Theorem 23.4.7

Equal chords mark out equal arcs.

This basically means that if you have 2 chords of the same length, the sector of the circle they mark out will be equal.

Theorem 23.4.9 · Right Angle Tangency Point

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

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