Properties of Angles
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Before working through the information below you may wish to review this lesson on measuring angles as well as taking a look at the lesson on adding and subtracting angles.
This lesson will provide information and guidance on:
Complementary Angles are those which add together to make 90°.
|∠ABD + ∠DBC = 90°|
These two angles are complementary because they add together to make 90°.
60° + 30° = 90°
These two angles are also complementary.
15° + 75 ° = 90°
The examples above all show two angles that are complementary. Notice that the angles do not have to be adjacent to be complementary. If they are adjacent then they form a right angle.
Supplementary Angles add together to make 180°
125° + 55° = 180°
The two angles shown above are supplementary to each other. They add together to give 180°. They can be said supplement each other. Note that, as with complementary angles, they do not need to be adjacent to each other.
When to lines intersect they create four angles. Each angle is opposite to another and form a pair of what are called opposite angles.
Angles a and c are opposite angles.
Angles b and d are opposite angles
Opposite angles are equal. The two 130° angles are opposite as are the two 50° angles.
Opposite angles are sometimes called vertical angles or vertically opposite angles.
Corresponding and Alternate Angles
The example below shows two parallel lines and a transversal (a line that cross two or more other lines). This results in eight angles. Each of these angles has a corresponding angle. Looking at the two intersections, the angles that are in the same relative (or corresponding) positions are called corresponding angles.
Since the two lines are parallel, the corresponding angles are equal.
a and e are corresponding angles
b and f are corresponding angles
c and g are corresponding angles
d and h are corresponding angles
As Shown below, there are also two pairs of alternate interior angles and two pairs of alternate exterior angles. Notice how the interior angles are in between the two parallel lines and the exterior angles are to the outside.
a and g are alternate exterior angles
b and h are alternate exterior angles
c and e are alternate interior angles
d and f are alternate interior angles
Since the two lines are parallel, the alternate angles shown above are equal.
Angle Relationship Worksheet
Have your children try the worksheet below that has questions on angle relationships. After completing it your children will be ready to review the lesson on finding missing angles.
After reviewing the lessons above you will be ready to read through the information below on angles and their relationships with your children. Discuss these as you go and, when you are ready, try the angle relationships worksheet.
Parallel Lines - lines that are equidistant from each other and never intersect.
Transversal - a line that intersects two or more other lines.
Adjacent Angles - angles that share a common side and that have a common vertex.
Types of Angle Pairs
Adjacent angles: two angles with a common vertex, sharing a common side and no overlap.
Angles ∠1 and ∠2 are adjacent.
Complementary angles: two angles, the sum of whose measures is 90°.
Angles ∠1 and ∠2 are complementary.
Complementary are these angles too(their sum is 90°):
Supplementary angles: two angles, the sum of whose measures is 180°.
Angles ∠1 and ∠2 are supplementary.
Angle pairs formed by parallel lines cut by a transversal
When two parallel lines are given in a figure, there are two main areas: the interior and the exterior.
When two parallel lines are cut by a third line, the third line is called the transversal. In the example below, eight angles are formed when parallel lines m and n are cut by a transversal line, t.
There are several special pairs of angles formed from this figure. Some pairs have already been reviewed:
∠1 and ∠4
∠2 and ∠3
∠5 and ∠8
∠6 and ∠7
Recall that all pairs of vertical angles are congruent.
∠1 and ∠2
∠2 and ∠4
∠3 and ∠4
∠1 and ∠3
∠5 and ∠6
∠6 and ∠8
∠7 and ∠8
∠5 and ∠7
Recall that supplementary angles are angles whose angle measure adds up to 180°. All of these supplementary pairs are linear pairs. There are other supplementary pairs described in the shortcut later in this section. There are three other special pairs of angles. These pairs are congruent pairs.
Alternate interior angles two angles in the interior of the parallel lines, and on opposite (alternate) sides of the transversal. Alternate interior angles are non-adjacent and congruent.
Alternate exterior angles two angles in the exterior of the parallel lines, and on opposite (alternate) sides of the transversal. Alternate exterior angles are non-adjacent and congruent.
Corresponding angles two angles, one in the interior and one in the exterior, that are on the same side of the transversal. Corresponding angles are non-adjacent and congruent.
Use the following diagram of parallel lines cut by a transversal to answer the example problems.
What is the measure of ∠8?
The angle marked with measure 53° and ∠8 are alternate exterior angles. They are in the exterior, on opposite sides of the transversal. Because they are congruent, the measure of ∠8 = 53°.
What is the measure of ∠7?
∠8 and ∠7 are a linear pair; they are supplementary. Their measures add up to 180°. Therefore, ∠7 = 180° – 53° = 127°.
1. When a transversal cuts parallel lines, all of the acute angles formed are congruent, and all of the obtuse angles formed are congruent.
In the figure above ∠1, ∠4, ∠5, and ∠7 are all acute angles. They are all congruent to each other. ∠1 ≅ ∠4 are vertical angles. ∠4 ≅ ∠5 are alternate interior angles, and ∠5 ≅ ∠7 are vertical angles. The same reasoning applies to the obtuse angles in the figure: ∠2, ∠3, ∠6, and ∠8 are all congruent to each other.
2. When parallel lines are cut by a transversal line, any one acute angle formed and any one obtuse angle formed are supplementary.
From the figure, you can see that ∠3 and ∠4 are supplementary because they are a linear pair.
Notice also that ∠3 ≅ ∠7, since they are corresponding angles. Therefore, you can substitute ∠7 for ∠3 and know that ∠7 and ∠4 are supplementary.
In the following figure, there are two parallel lines cut by a transversal. Which marked angle is supplementary to ∠1?
The angle supplementary to ∠1 is ∠6. ∠1 is an obtuse angle, and any one acute angle, paired with any obtuse angle are supplementary angles. This is the only angle marked that is acute.
Angles - Problems with Solutions
Types of angles
Parallel lines cut by a transversal Test