How can we use congruent angles?
Shapes are considered congruent if they have the exact same shape with equal side lengths and equal angle measures. Look at the diagrams below for an example of two congruent shapes.
As you can see, the shapes are the same size and are equal in lengths and angle measures. When a transformation, such as a translation, rotation, or reflection is done on a shape, the two shapes are considered congruent.
The symbol below is used to represent congruence. We write this between naming of a side length, angle,, or shape to show that two or more things are congruent.
In parallel lines, there are a number of angle pairs. Give the diagram below:
All angles that are colored in blue are congruent. All angles that are colored in red are congruent. There are many different angle pair relationships found within. A few examples of these are as follows.
Angle 1 and Angle 3 are known as corresponding angles. This is the same for angles 2 and 4, 5 and 7, and 6 and 8.
Angle 1 and Angle 8 are known as alternating exterior angles. This is the same for angles 5 and 4.
Angle 2 and Angle 7 are known as alternating interior angles. This is the same for angles 6 and 3.
Angles 1 and 6 are known as vertical angles. The same is for angles 5 and 2, 3 and 8, and 7 and 4.
All of these angle pairs have angles whose measures are equal and are therefore known as congruent angles.
Shapes that are similar, their sides are proportional and their angles are congruent. When given two triangles, you can conclude that they are similar based upon whether they have at least two angles that are congruent. There is a postulate known as the Angle-Angle Postulate which states if two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. Knowing that they are similar, you can then use proportions to solve for side lengths or other algebraic expressions.
Sample Math Problems
1. In the figure, the measure of angle 3 is 102 degrees with m||n and p||q. Find the measure of the following angles:
a) Angle 5
Solution:
Angle 3 and Angle 5 are alternate interior angles so their measures are equal. Angle 5 is also 102 degrees.
Angle 3 and Angle 5 are alternate interior angles so their measures are equal. Angle 5 is also 102 degrees.
b) Angle 11
Solution:
Angle 11 is a corresponding angle to angle 3,
therefore angle 11 is also 102 degrees.
c) Angle 6
Solution:
Solution:
Angle 3 and Angle 2 are supplementary, so angle 2 is 78 degrees. Then Angles 2 and
Angle 6 are corresponding angles, therefore angle 6 is 78 degrees
2.
Find x. Then find the measure of angle D.
Solution: Angle D is congruent to angle B. Therefore we can set their values equal to each other.
2x + 7 = x + 20
x + 7 = 20
x = 13
Angle D = 2x + 7 = 2(13) + 7 = 26 + 7 = 33 degrees
