∠Angle RelationshipsUnit 3 · Geometry

Exercise 12

Remember · Understand · Apply · Problem solving. Every answer can be revealed after you have written your own.

Section 1

Complete each of the following

  1. 1If two straight lines intersect, then each two vertically opposite angles are …

  2. 2The sum of the measures of the accumulative angles at a point equals …

  3. 3Two adjacent angles on a straight line always add up to …

  4. 4If two vertically opposite angles are complementary, then the measure of each is …

  5. 5Three straight lines meeting at one point form … pairs of vertically opposite angles.

Section 2

Find what is required below each figure

(3x + 12)°84°BCADM
1 — AB ∩ CD = {M}. Find x.
(5x − 8)°(3x + 26)°BCADM
2 — AB ∩ CD = {M}. Find x.
38°x°BCADM
3 — Two lines meet at E. Find x.
60°50°40°(x − 7)°ABCDEM
4 — Angles at M. Find x.
4x°80°3x°ABCDM
5 — Angles at M. Find x.
(2x)°(x + 28)°BCADM
6 — Vertically opposite angles. Find the measure of one.

Section 3

Multiple choice questions

Choose the correct answer, then check all ten at once.

  1. 1. The angle of measure 70° is vertically opposite to an angle of measure …

  2. 2. The sum of the measures of the angles around a point equals the measure of …

  3. 3. The sum of the measures of 4 angles around a point is … the sum of the measures of 5 angles around a point.

  4. 4. If two vertically opposite angles are complementary, then the measure of each is …

  5. 5. AB ∩ CD = {M}. If m(∠AMD) = 60° and m(∠BMC) = 6x°, then x = …

  6. 6. Around a point the angles are 120°, 7x°, 5x° and 4x°. The value of x is …

  7. 7. Two straight lines intersect. One angle measures 108°. The angle adjacent to it measures …

  8. 8. Three different straight lines intersect at one point. The number of pairs of vertically opposite angles equals …

  9. 9. If the ratio among the measures of 3 angles around a point is 3 : 4 : 5, then the greatest angle measures …

  10. 10. Around a point there are exactly three angles: 90°, (90 − x)° and (80 + x)°. Wait — the third angle ∠AMB measures …

Section 4

Creative thinking

Longer problems that combine bisectors, ratios and several relationships.

  1. 1. AC ∩ BD = {M}, ray MX bisects ∠AMB and m(∠CMD) = 140°. Find m(∠DMX).

  2. 2. Around the point M: m(∠BMC) = 80°, m(∠CMD) = 110°, m(∠DME) = 90° and m(∠AMB) : m(∠AME) = 2 : 3. Find m(∠AME) and m(∠AMB).

  3. 3. The angles between the blades of a pair of scissors measure (3x − 18)° and (2x + 12)° and they are vertically opposite. Find the measure of each angle.

  4. 4. m(∠BMC) = 2 m(∠AMB), m(∠AMB) = 48° and m(∠DMC) = 115° are angles at the point M. Find m(∠AMD).