Angles in a Triangle

Angle and Triangle Relationships

Yes, angles and triangles date each other. This is a story about them.

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Pre-Algebra

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Take notes as we go!​

Adjacent Angles

• Two angles that share a common side and vertex

• next to each other (next door neighbors!)​

• ∠a is adjacent to ∠b

Multiple Choice

Angle a and angle b are ____ angles.

• complementary

• supplementary

• adjacent

Complementary Angles

• Two angles add up to 90 degrees

• Both angles are acute (less than 90 degrees)

• They do NOT have to be adjacent (or next to each other). Any two angles that add up to 90 degrees are complementary.

Multiple Choice

Because these two angles add up to 90 degrees, we know they are _____.

• supplementary angles.

• complementary angles.

• adjacent angles.

• interior angles.

Multiple Choice

What is the value of x?

• 61 degrees

• 29 degrees

• 51 degrees

• 59 degrees

Supplementary Angles

• Two angles that add up to 180 degrees

• One acute angle (less than 90 degrees) and one obtuse angle (more than 90 degrees)

• They do NOT have to be adjacent (or next to each other). Any two angles that add up to 180 degrees are supplementary.

Multiple Choice

Because these two angles add up to 180 degrees, we know they are ____.

• alternate angles

• supplementary angles

• complementary angles

• adajenct angles

Multiple Choice

Supplementary angles together make a ____.

• 270 degree angle

• right angle

• straight line

• 70 degree angle

Multiple Choice

What is the value of x?

• 133 degrees

• 143 degrees

• 58 degrees

• 70 degrees

Linear Pair

• Two adjacent angles that form a straight line

• Are always supplementary

• Example:​ If two angles form a linear pair, and one angle is 80 degrees, then we know the other angle is 100 degrees.

• (180 - 80 = 100)

Multiple Choice

Two angles form a linear pair, if one is 63 degrees, what is the measure of the other angle?

• 127 degrees

• 107 degrees

• 117 degrees

• 32 degrees

Vertical Angles

• 2 angles that are directly across from each other when 2 lines cross.

• These vertical angles are congruent (equal) in their angle measurement

• ∠Y ≅ ∠Y and ∠X ≅ ∠X

• Example: if one vertical angle is 111⁰​, then the other is equal or x=111⁰​

Multiple Choice

In the picture, which angle is the vertical angle to angle 8?

• ∠8 and ∠7

• ∠8 and ∠6

• ∠8 and ∠5

Multiple Choice

Which two angles do we know are congruent because they are vertical angles?

• ∠16 ≅ ∠14

• ∠16 ≅ ∠13

• ∠16 ≅ ∠15

• ∠16 ≅ ∠10

Multiple Choice

What is the value of x?

• 90 $$\degree$$  

• 180 $$\degree$$  

• 99 $$\degree$$  

• 101 $$\degree$$  

Exterior & Remote Interior Angles

• The angle outside the triangle, when a line is extended from the triangle is exterior angle (∠d)

• The 2 opposite interior angles are also known as remote interior angles. (∠a and ∠b)

• Exterior Angle Theorem says the sum of the 2 opposite interior angles equals the exterior angle. (∠a + ∠b = ∠d)

Multiple Choice

What is the measurement of the exterior angle?

• 145

• 45

• 80

• 65

Multiple Choice

Because of the Exterior Angle Theorem we know that the triangle in the picture:

• 145 = 80 + x

• 145 = 80 - x

• 145 + 80 = x

Adjacent Interior and Exterior Angles

• The angle outside the triangle, when a line is extended from the triangle is exterior angle. (angle d)

• Adjacent Interior Angle is next to the exterior angle. (angle c)

• The sum of the adjacent interior angle and the exterior angle equal 180 degrees (∠c + ∠d = 180)

Multiple Choice

Which statement is true in the picture?

• ∠b + ∠d = 90

• ∠b + ∠d = 180

Multiple Choice

Which angle is the adjacent interior angle?

• ∠b

• ∠d

• ∠a

• ∠c

Parallel Lines and Transversal

• Parallel Lines are two lines that never intersect. (The symbol ∥ means "parallel") MA ∥ ZE

• A Transversal is a line cutting across other lines (OH is the transversal)

Corresponding Angles

• If the two angles are in the same position in each intersection, they are called corresponding angles.

• In the picture, the corresponding angles are:

• ∠B and ∠G;

• ∠D and ∠K;

• ∠C and ∠J;

• ∠F and ∠L (the two circled angles)

Multiple Choice

Which angles are corresponding angles?

• ∠8 and ∠4

• ∠7 and ∠4

• ∠5 and ∠4

Alternate Interior Angles

• The Alternate Interior Angles are located INSIDE the two parallel lines.

• The Alternate Interior Angles are equal (or congruent).

• ∠D ≅ ∠J (angle D is congruent to angle J)

Multiple Choice

Which angles are alternate interior angles?

• ∠8 and ∠4

• ∠7 and ∠4

• ∠5 and ∠4

Alternate Exterior Angles

• Alternate Exterior Angles are on the OUTSIDE of the parallel lines.

• Alternate Exterior Angles are equal (or congruent)

• ∠B ≅ ∠L (angle B is congruent to angle L)

Multiple Choice

Which angles are alternate exterior angles?

• ∠8 and ∠4

• ∠7 and ∠1

• ∠8 and ∠1

Supplementary Angles

• Supplementary Angles are created by the transversal.

• Supplementary angles add up to 180 degrees

• ∠B + ∠C = 180 (angle B plus angle C equals 180 degrees)

Consecutive Interior Angles

• If two parallel lines are cut by a transversal, the consecutive interior angles are supplementary.

• ∠F + ∠J = 180 (angle F plus angle J equals 180 degrees)

• ∠D + ∠G = 180 (angle D plus angle G equals 180 degrees)

Multiple Choice

Which angles are consecutive interior angles?

• ∠8 and ∠7

• ∠3 and ∠4

Consecutive Exterior Angles

• If two parallel lines are cut by a transversal, the consecutive exterior angles are supplementary.

• ∠B + ∠C = 180 (angle F plus angle J equals 180 degrees)

• ∠K + ∠L = 180 (angle D plus angle G equals 180 degrees)

Multiple Choice

Which angles are consecutive exterior angles?

• ∠1 and ∠2

• ∠3 and ∠4

• ∠7 and ∠4

Similar Triangles

• Two triangles are similar if all their angles are equal and corresponding sides are the same ratio.

• AC and ZX are 21:14 = 3:2 ratio

• AB and YX are 15:10 = 3:2 ratio

Similar Triangles with Similar Slope

• You can watch a short video to explain by clicking at this link: https://drive.google.com/file/d/14DSyovnzfMVx4Y5Sc7Zs52pMfV-GgQNo/view

• In the picture, the rise is 4 and the run is 3. A similar triangle can be created by multiplying both by the same number.

• A similar triangle could have a rise of 8 (4x2) and a run of 6 (3x2)

Multiple Choice

If you were to create a similar triangle with a rise of 12, the run would be...

• 8

• 9

• 10

Now, practice some test questions!