Learn – Geometry – Angles

Axioms and Theorems

An axiom is a statement that is taken for granted to be true.

A theorem is a statement justified (proven) to be true.

Intersecting lines

Exercises

Problem set

  1. In the following figure, what is the measure of \angle ACD?
  2. In the following figure, what are the measures of \angle ACD and \angle ACE?
  3. In the following figure, what are the measures of \angle BCE, \angle ACE and \angle ACD?
  4. In the following figure, AB is a straight line segment. What is the measure of \angle DCE?
  5. In the following figure, \angle BAC : \angle CAD = 3:2, \angle CAD : \angle DAE = 5:3, \angle DAE : \angle EAF = 4:7. What is the measure of \angle EAC?

Parallel lines

Exercises

Problem set

  1. In the following figure, AB \parallel CD. What are the measures of \angle DHF and \angle CHF?
  2. In the following figure, AB \parallel CD. What are the measures of \angle AGH and \angle HGB?
  3. In the following figure, AB\parallel CD. What is the value of x?
  4. In the following figure, \overleftrightarrow{AB}\parallel \overleftrightarrow{CD}. What is the measure of \angle EFG?
  5. In the following figure, \overleftrightarrow{AB}\parallel\overleftrightarrow{CD}. What is the measure of \angle AHP?

Problem set

  1. In the following figure, \overleftrightarrow{AB}\parallel \overleftrightarrow{CD}. What is the measure of \angle ARG?
  2. In the following figure, \overleftrightarrow{AB}\parallel \overleftrightarrow{CD} and FP=PG. What is the measure of \angle AHP?
  3. In the following figure, AB\parallel CD, and PQ = PE. What is the measure of \angle PSR?

Problem set

  1. In the following figure, AB\parallel CD, EF=GH and HI is a straight line segment. What is the measure of \angle EGI?
  2. In the following figure, AB\parallel CD, EF=GH and HI is a straight line segment. What is the measure of \angle GIJ?
  3. In the following figure, ABDC is a trapezoid with AB=CD. What is the measure of \angle BDC?
  4. In the following figure, AB\parallel CD, EF=GH=HI. What is the measure of \angle CEF?
  5. In the following figure, AB\parallel CD, EF=GH=HI, and FJ and IK are straight line segments. What is the measure of \angle JKG?

Triangles

Exercises

Problem set

  1. In the following figure, what is the measure of \angle ACB?
  2. In the following figure, what is the measure of \angle ACB?
  3. In the following figure, what is the measure of \angle RPQ?
  4. In the following figure, what is the measure of \angle ABC?
  5. In the following figure, what is the measure of \angle CBA?
  6. In the following figure, what is the measure of \angle ACB?

Problem set

  1. In the following figure, what is the measure of \angle BFE?
  2. In the following figure, what is the measure of \angle QPR?
  3. In the following figure, what is the measure of \angle BCA?
  4. In the following figure, what is the measure of \angle BAD?
  5. In the following figure, AB, CD and EF are three straight line segments. What is the measure of \angle QPC?
  6. In the following figure, AD is a straight line segment. what is the measure of \angle DBC?

Problem set

  1. In the following figure, \angle BAC = \angle ADC and BD is a straight line segment. What is the measure of \angle DAC?
  2. In the following figure, AB is a straight line segment. What is the value of x?
  3. In the following figure, AB is a straight line segment. And, AD=AC and CD=DB. What is the measure of \angle DBC?
  4. In the following figure, AB is a straight line segment. What is the measure of \angle DAC?
  5. In the following figure, AD is a straight line segment. What is the length of segment AC?

Problem set

  1. In the following figure, BD is a straight line segment. What is the measure of \angle CAD?
  2. In the following figure, ABC is a right triangle. And, \angle BAE = \angle EAC and \angle DCB = \angle DCA. What is the measure of \angle DFE?
  3. In the above figure, if \angle DCA = 20^\circ, what is the measure of FEC?

Polygons

Exercises

Problem set

  1. In the following, AF is a straight line segment. What is the measure of angle \angle DEF?

  2. What is the measure of each of the interior angles of a regular pentagon?
  3. What is the measure of each of the interior angles of a regular hexagon?
  4. The measure of the interior angle of a regular polygon is 120^\circ. How many sides does the polygon have?
  5. The measure of the interior angle of a regular polygon is 140^\circ. How many sides does the polygon have?

Problem set

  1. In the following figure, ABCDE is a regular polygon. What is the measure of \angle CAE?
  2. In an octagon, the interior angles are in the ratio 1:2:3:4:5:6:7:8. What is the measure of the smallest angle?
  3. One angle of a hexagon is 32^\circ. Remaining angles of the hexagon are in the ratio 2:2:3:4:5. What is the biggest angle of the hexagon?
  4. Two of the interior angles of a hexagon are 90^\circ and 100^\circ. Of the remaining four angles, the measures of the first and the second are in the ratio 4:3, the measures of the second and the third are also in the ratio 4:3 and the measures of the third and the fourth are also in the ratio 4:3. What is the measure of the biggest angle?

Radian measure

Definition of 1 radian

Exercises

Problem set
  1. An arc on the circumference of a circle subtends (makes) an angle of 40^\circ at the center of the circle, and the length of the arc is 5 units. Another arc on the circumference of the same circle subtends an angle of 80^\circ. What is the length of the second arc?
  2. An arc on the circumference of a circle subtends (makes) an angle of 50^\circ at the center of the circle, and the length of the arc is 4 units. Another arc on the circumference of the same circle subtends an angle of 25^\circ. What is the length of the second arc?
  3. An arc on the circumference of a circle subtends (makes) an angle of 40^\circ at the center of the circle, and the length of the arc is 10 units. Another arc on the circumference of the same circle subtends an angle of 60^\circ. What is the length of the second arc?
  4. An arc on the circumference of a circle subtends (makes) an angle of 60^\circ at the center of the circle, and the length of the arc is 4 units. How much of an angle does a second arc of length 2 units on the circumference of the same circle subtend at the center of the circle?
  5. An arc on the circumference of a circle subtends (makes) an angle of 90^\circ at the center of the circle, and the length of the arc is 6 units. How much of an angle does a second another arc of length 4 units on the circumference of the same circle subtend at the center of the circle?
Problem set
  1. A circle has radius 4 units. An arc on the circumference of the circle subtends(makes) an angle of 1 radian at the center of the circle. What is the length of the arc?
  2. A circle has radius 1cm. What is the measure of the angle in radians that is subtended (made) by an arc of length 2cm at the center of the circle?
  3. A circle has radius 5 units. An arc on the circumference of the circle subtends(makes) an angle of 3 radians at the center of the circle. What is the length of the arc?
  4. A circle has radius 3 units. An arc on the circumference of the circle subtends(makes) an angle of 4 radians at the center of the circle. What is the length of the arc?
  5. A circle has radius 2cm. What is the measure of the angle in radians that is subtended (made) by an arc of length 8cm at the center of the circle?
  6. A circle has radius 12cm. What is the measure of the angle in radians that is subtended (made) by an arc of length 18cm at the center of the circle?
  7. A circle has radius 4m. What is the measure of the angle in radians that is subtended (made) by an arc of length 10m at the center of the circle?
  8. A circle has radius 5m. What is the measure of the angle in radians that is subtended (made) by an arc of length 13m at the center of the circle?
  9. A circle has radius 12cm. What is the length of the arc that subtends (makes) an angle of 0.25 radians at the center?
  10. In a circle, an arc of length 24cm subtends (makes) an angle of 0.5 radians at the center of the circle. What is the radius of the circle?

Conversion from degrees to radians

Exercises

Problem set

Express each of the following in radian measure.

  1. 90^\circ
  2. 180^\circ
  3. 270^\circ
  4. 360^\circ
  5. 60^\circ
  6. 30^\circ
  7. 45^\circ
Problem set

Express each of the following in radian measure.

  1. 15^\circ
  2. 120^\circ
  3. 135^\circ
  4. 150^\circ
  5. 270^\circ
Problem set

Express each of the following in radian measure.

  1. 210^\circ
  2. 240^\circ
  3. 360^\circ
  4. 300^\circ
  5. 315^\circ
Problem set

Express each of the following angles as 360^\circ \pm \theta, where 0^\circ \le \theta \le 180^\circ. For example, 315^\circ = 360^\circ - 45^\circ. Then, use such expressions to convert each of the given angles into radian measure.

  1. 330^\circ
  2. 390^\circ
  3. 315^\circ
  4. 405^\circ
  5. 420^\circ
Problem set

Express each of the following in radian measure.

  1. -75^\circ
  2. -135^\circ
  3. -210^\circ
Problem set

Express each of the following angles as -360^\circ \pm \theta, where 0^\circ \le \theta \le 180^\circ. For example, -315^\circ = -360^\circ + 45^\circ. Then, use such expressions to convert each of the given angles into radian measure.

  1. -330^\circ
  2. -390^\circ
  3. -510^\circ
  4. -495^\circ
Problem set

Express each of the following angles as M \pm \theta, where M is either a multiple of 360^\circ and 0^\circ \le \theta \le 180^\circ. For example, 660^\circ = 720^\circ - 60^\circ = 2\times 360^\circ - 60^\circ. Then, use such expressions to convert each of the given angles into radian measure.

  1. 1500^\circ
  2. -1110^\circ
  3. -855^\circ
  4. 690^\circ
  5. -1020^\circ
  6. 1410^\circ

Conversion from radians to degrees

Exercises

Problem set

Express each of the following in degrees.

  1. \frac{\pi}{3}
  2. 2\pi
  3. \frac{\pi}{4}
  4. \frac{\pi}{6}
  5. \frac{\pi}{12}
Problem set

Express each of the following in degrees.

  1. \frac{2\pi}{3}
  2. \frac{3\pi}{4}
  3. \frac{5\pi}{6}
  4. \frac{3\pi}{2}
  5. \frac{4\pi}{3}
Problem set

Express each of the following angles as M \pm \theta, where M is a multiple of 2\pi and 0\le \theta \le \pi. For example, \frac{14\pi}{3} = \frac{12\pi}{3} + \frac{2\pi}{3} = 4\pi + \frac{2\pi}{3}. Then, use such expressions to convert each of the given angles into degrees.

  1. \frac{13\pi}{3}
  2. -\frac{25\pi}{6}
  3. -\frac{140\pi}{60}
  4. \frac{15\pi}{4}
  5. -\frac{17\pi}{6}
  6. -\frac{14\pi}{3}
  7. \frac{75\pi}{20}

Application of concept

Exercises

Problem set
  1. A circle has radius 10 units. An arc on the circumference of the circle subtends (makes) an angle of 90^\circ at the center of the circle. What is the length of the arc?
  2. A circle has radius 10 units. An arc on the circumference of the circle subtends (makes) an angle of 60^\circ at the center of the circle. What is the length of the arc?
  3. A circle has radius 10 units. An arc on the circumference of the circle subtends (makes) an angle of 9^\circ at the center of the circle. What is the length of the arc?
  4. Two circles are drawn. First circle has radius r units, and the second circle has radius 3r units. An arc of length a units on the circumference of the first circle subtends (makes) an angle of \theta at the center of the circle. What is the length of the arc on the circumference of the second circle that subtends an angle of \theta at the center of the second circle?
  5. Two circles have radii in the ratio 2:1. An arc of length 6cm in the smaller circle subtends (makes) an angle of 2^c at the center. What would be the length of the arc in the bigger circle that subtends the same angle?
  6. Two circles have radii in the ratio 4.5:1. An arc in the bigger circle has has length that is 9 times the length of an arc in the smaller circle. What is the ratio of angles subtended (made) by the two arcs at the respective centers?
  7. Two circles have radii in the ratio 4.5:1. An arc in the bigger circle has has length that is 1.5 times the length of an arc in the smaller circle. What is the ratio of angles subtended (made) by the two arcs at the respective centers?
Problem set
  1. In an isosceles triangle, each of the equal angles is \frac{\pi}{33} radians. What is the measure of the third angle in radians?
  2. Two radii are drawn for a circle to have an arc of length \frac{2\pi}{3} feet between them. The angle subtended by the two radii is 60^\circ. What is the radius of the circle?
  3. ABCDEFGH is a regular octagon. What is the angular measure of each of its interior angles in radians?