Identify the pairs of congruent corresponding angles and the corresponding sides

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3. Corresponding Sides/angles

Angles, sides in the same 'spot'

What are corresponding sides and angles?

Corresponding sides and angles are a pair of matching angles or sides that are in the same spot in two different shapes. Look at the pictures below to see what corresponding sides and angles look like.

Note:

These shapes must either be similar or congruent.

Example 1

In $$\triangle \red{A}BC $$ and $$\triangle \red{X}YZ $$,
$$\angle A$$ corresponds with $$\angle X$$.

In quadrilaterals $$\red{JK}LM$$ and $$\red{RS}TU$$,
$$ \overline {JK} $$ corresponds with $$ \overline{RS} $$ .

Example 2

In quadrilaterals $$ABC\red{D}E $$ and $$HIJ\red{K}L $$,
$$\angle D$$ corresponds with $$\angle K$$.

In quadrilaterals $$A\red{BC}DE $$ and $$H\red{IJ}KL $$,
$$ \overline {BC} $$ corresponds with $$ \overline {IJ} $$.

Interactive Demonstration

What if the shapes are rotated around?

Orientation does not affect corresponding sides/angles. It only makes it harder for us to see which sides/angles correspond.

The two triangles below are congruent and their corresponding sides are color coded. Try pausing then rotating the left hand triangle. Notice that as the triangle moves around it's not always as easy to see which sides go with which. (Imagine if they were not color coded!).

Practice Problems

Problem 1

If $$\triangle ABC $$ and $$ \triangle UYT$$ are similar triangles, then what sides/angles correspond with:

AB	Follow the letters the original shapes: $$\triangle \red{AB}C $$ and $$ \triangle \red{UY}T $$. Answer:UY
$$ \angle BCA $$	Follow the letters the original shapes: $$\triangle ABC $$ and $$ \triangle UYT $$. Answer: $$ \angle YTU $$
TU	Follow the letters the original shapes: $$\triangle\red{A}B\red{C} $$ and $$ \triangle \red{U} Y \red{T} $$. Answer: CA
$$ \angle TUY $$	Follow the letters the original shapes: $$\triangle ABC $$ and $$ \triangle UYT $$. Answer: $$ \angle CAB$$

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Congruent Angle Overview

Congruent angles are frequently used in the world of architecture, construction, design, and art. Congruent angles have the same angle measure. For example, a regular pentagon has five sides and five angles, and each angle is 108 degrees. Regardless of the size or scale of a regular polygon, the angles will always be congruent.

Congruent Angle Sample Questions

There are many rules that allow us to determine whether angles are congruent or not. For example, if two triangles are similar, their corresponding angles will be congruent. This means that the angles that are in the same matching position will have the same angle.

Another common test for angle congruence requires a set of parallel lines and a transversal line that slices through the set of parallel lines. For example, lines a and b are parallel, and line l is a transversal that slices through the parallel lines. When this situation occurs, a handful of congruent angles are formed. There are four main types of congruent angles formed in this scenario: Alternate Interior Angles, Alternate Exterior Angles, Corresponding Angles, and Vertical Angles.

Similarly, Alternate Exterior Angles are located on the outside of the parallel lines, and on alternate sides of the transversal. ∠5 and ∠1 are congruent, as well as ∠4 and ∠8.

Corresponding Angles are located on the same side of the transversal, and in a similar matching location. For example, ∠4 and ∠6 are corresponding angles, therefore they are congruent. Other pairs of corresponding angles include ∠3 and ∠5, ∠1 and ∠7, and ∠2 and ∠8.

Vertical Angles are formed by angles that are opposite of each other. For example, ∠1 and ∠3, ∠7 and ∠5, ∠4 and ∠2, ∠6 and ∠8 are all pairs of congruent angles. Vertical angles, or opposite angles, are commonly used as a proof of congruence.

Another category of congruent angles revolves around triangle congruence. Triangle congruence rules are used to prove if two triangles are congruent or not. These rules take into consideration the side lengths and angles of triangles in order to determine congruence. Four criteria are used to determine triangle congruence, and they are conveniently named.

For example:
S-S-S refers to two triangles that have all side lengths the same. If this is true, then all the corresponding angle measures will be congruent as well.

S-A-S refers to two triangles that have two congruent sides, with one congruent angle in between. If this is true, then all the corresponding angles will be congruent.

Similarly, A-S-A tells us that two triangles have two congruent angles, with one congruent side length in between. Again, if this is true, then all the corresponding angles will be congruent.

Lastly, A-A-S refers to two triangles that have two corresponding congruent angles, with a corresponding congruent side length. This tells us that all the corresponding angles will be congruent.

Congruent angles are commonly used in the study of Geometry, and in many real-world occupations. Construction workers, engineers, builders, and artists use congruent angles on a regular basis. Determining whether angles are congruent is an important skill that helps lay the foundation for the study of Geometry.

Here are a few sample questions going over congruent angles.

Question #1:

 
Angles 1 and 2 are corresponding angles. If the measure of Angle 2 is 67°, what is the measure of Angle 1?

Show Answer

Answer:

When two parallel lines are cut by a transversal, the angles that are on the same side of the transversal and in matching corners, will be congruent. Angles 1 and 2 are congruent angles, so both have an angle measure of 67°.

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Question #2:

 
Line r is a transversal that crosses through the two parallel lines s and t. List all angles that are congruent to Angle 6.

∠8, ∠3, and ∠2

∠8, ∠4, and ∠2

∠2

∠1, ∠7, and ∠2

Show Answer

Answer:

∠8 is congruent to ∠6 because they are vertical angles, or opposite angles.
∠2 is congruent to ∠6 because they are corresponding angles (same side of the transversal and in matching corners).
∠4 is congruent to ∠6 because they are alternate interior angles (alternate sides of the transversal, and between the two parallel lines).

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Question #3:

 
If the following irregular quadrilaterals are congruent, Angle C must be congruent to what other angle?

Show Answer

Answer:

Corresponding angles are congruent for polygons that are congruent. ∠C is congruent to ∠G
∠D is congruent to ∠H
∠A is congruent to ∠E
∠B is congruent to ∠F

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Question #4:

 
The city of Seattle is building a walking path that crosses over a pair of railroad tracks. The walking path is represented by the transversal t in the image below. The railroad tracks are represented by the parallel lines l and m. If the city wants to have the walking path cross the tracks at a 135° angle (Angle 1), what will the values of Angles 2, 3, and 4 be?

∠2 = 45° ∠3 = 135° ∠4 = 45°

∠2 = 45° ∠3 = 45° ∠4 = 135°

∠2 = 145° ∠3 = 45° ∠4 = 45°

∠2 = 180° ∠3 = 45° ∠4 = 45°

Show Answer

Answer:

∠1 and ∠4 are congruent because they are vertical angles. If ∠1 equals 135°, then ∠2 must be equal to 45° because their sum needs to be 180° in order to form a straight line. Now that we know ∠2 equals 45°, we also know that ∠3 equals 45° because they are vertical angles.

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Question #5:

 
Kelcy has a rectangular garden that she wants to divide equally into two sections diagonally. One section will be for carrots and the other section will be for kale. She separates the garden into two triangular pieces similar to the image below. If the measure of ∠DCA is 40° what is the measure of ∠CAB?

Show Answer

Answer:

If the line AC forms a transversal through the parallel lines DC and AB, then the angles DCA and CAB will be congruent. Angle DCA and Angle CAB form alternate interior angles, so the measure of Angle CAB will be 40°.

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Return to Math Sample Questions

What are pairs of congruent angles?