Adjacent and Vertical Angles

In this activity, you will learn about angle pairs that are formed when two lines intersect.

  1. In the Gizmotm, under Conditions, select None. Be sure that Adjacent is selected.
    1. Notice angleAOB and angleBOC . What do you observe about the vertex of each angle? What else do these angles share?
    2. Angles that share a vertex and have a common side but no common interior points are called adjacent angles. Drag A, B, and C to see other pairs of adjacent angles. On paper, make a sketch of two adjacent angles.
  2. Under Conditions, select Vertical angles.
    1. Vertical angles are pairs of angles that are formed by intersecting lines and are opposite each other. Name two pairs of vertical angles in the diagram shown in the Gizmo.
    2. Click on Click to measure angles to turn on the Gizmo protractors. (For help using the protractors, click on Gizmo Help, below the Gizmo.) Use the protractors to measure angleAOB , angleBOC , angleCOD , and angleAOD . Record your data. Click on Show angle measures to check your work.
    3. You should have found that the pairs of vertical angles in this diagram are angleAOB and angleCOD , and angleBOC and angleAOD . What do you notice about the measures of these pairs of vertical angles? Make a conjecture about vertical angles.
    4. Drag A, B, C, and D to see more angles formed by two intersecting lines. Do vertical angles always seem to be congruent? Explain, and provide sketches of some of the angles you formed.

Complementary and Supplementary Angles

In this activity, you will learn about angle pairs whose measures add to 90° or 180° .

  1. Under Conditions, select Complementary to congruent angles. Be sure that Adjacent is selected. Adjust the angles by dragging points B and E.
    1. Two angles are complementary if their measures add up to 90° . Name two pairs of adjacent complementary angles in the diagram in the Gizmo. (You can also say that the angles are complements of each other.)
    2. Select Show angle measures. What is mangleDYE ? What is the sum of mangleDYE and mangleEYF ? Use this information to find mangleEYF . Then turn on a Gizmo protractor and measure angleEYF to check your answer.
    3. In general, if two angles are complementary and you know the measure of one of the angles, how can you find the measure of the second angle? Drag point E to form new angles and test your method. Write down one example that demonstrates your method. Does it always work?
    4. Complementary angles can be adjacent or nonadjacent. Select Nonadjacent in the Gizmo. Change the angles by dragging points E and F. Name two pairs of complementary angles. Turn on Show angle measures and use the protractor to confirm that the angles are complementary.
    5. It is given that angleAWB and angleEYF are congruent. What other pair of angles is congruent?
    6. Under Conditions, select Complementary to same angle. Name two pairs of complementary angles. (Hint: Look for overlapping right angles.)
    7. Turn on Show angle measures. Drag A, B, C, and D to form new angles. What two angles are always congruent?
    8. Use what you know about complementary angles to find mangleBOC . Then use the protractor to check your answer.
  2. Under Conditions, select Supplementary to same angle. With Adjacent selected, adjust the angles by dragging points A, B, C, and D.
    1. Two angles are supplementary if their measures add up to 180° . Name two different angles that are supplementary to angleBOC . (You can also say that the angles are supplements of each other.)
    2. Select Show angle measures. What is mangleAOB ? What is the sum of mangleAOB and mangleBOC ? Use this information to find mangleBOC . Then turn on a Gizmo protractor and measure angleBOC to check your answer.
    3. In general, if two angles are supplementary and you know the measure of one of the angles, how can you find the measure of the second angle? Drag any of the points to form new angles and test your method. Write down one example that demonstrates your method. Does it always work?
    4. Supplementary angles can be adjacent or nonadjacent. Select Nonadjacent in the Gizmo. Change the angles by dragging points A and B. Name the two pairs of supplementary angles.