Introduction to Angles and Triangles

Degrees: Measuring Angles We measure the size of an angle using degrees. Example: Here are some examples of angles and their degree measurements.

Acute Angles An acute angle is an angle measuring between 0 and 90 degrees. Example:                                                                

Obtuse Angles An obtuse angle is an angle measuring between 90 and 180 degrees. Example:                                                                  

Examples: Straight Angle A right angle is an angle measuring 180 degrees. Examples:                                       

Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees. Example: These two angles are supplementary.                                                                       

These two angles can be "pasted" together to form a straight line!                                                 

Example: Complementary Angles Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees. Example: These two angles are complementary.                                                                                

These two angles can be "pasted" together to form a right angle!                         

Review State whether the following are acute, right, or obtuse. acute 3. 5. 1. right obtuse ? 4. 2. acute ? obtuse

Complementary and Supplementary Find the missing angle. 1. Two angles are complementary. One measures 65 degrees. 2. Two angles are supplementary. One measures 140 degrees. Answer : 25 Answer : 40

Complementary and Supplementary Find the missing angle. You do not have a protractor. Use the clues in the pictures. 2. 1. x x 55 165 X=35 X=15

1. x x = 90 y z y = z = x = 2. x 110 y z y = z =

Vertical Angles are angles on each side of intersecting lines                     1. 90 x y z 90 90 and y are vertical angles x and z are vertical angles The vertical angles in this case are equal, will this always be true? 110 70 110 x y z 110 and y are vertical angles 2. x and z are vertical angles Vertical angles are always equal

Vertical Angles Find the missing angle. Use the clues in the pictures. x X=58 58

Can you find the missing angles? 20 90 D 70 G C 70 90 20 J H

Can you find these missing angles B C 68 A 52 G 60 D 60 52 68 F E

Can be classified by the angle measures Triangles Can be classified by the angle measures

Has three acute angles (less than 90 degrees) Acute Triangle Has three acute angles (less than 90 degrees)

Triangle with one obtuse angle (greater than 90 degrees) Obtuse Triangle Triangle with one obtuse angle (greater than 90 degrees)

Has one right angle (90 degree) Right Triangle Has one right angle (90 degree)

Can be classified by the number of congruent sides Triangles Can be classified by the number of congruent sides

Has no congruent sides (all angles, and sides are different) Scalene Triangle Has no congruent sides (all angles, and sides are different)

Has at least two congruent sides Isosceles Triangle Has at least two congruent sides

All three sides are congruent Equilateral Triangle All three sides are congruent

Which can only happen in a equalateral triangle Equiangular Triangle Triangle with 3 equal angles Which can only happen in a equalateral triangle

Triangles Cut any shape triangle out of a sheet of paper . Tear off the corners. Piece them together by having the corners touch. The corners form what type of angle?

The sum of the angles of a triangle is 180 degrees Triangles The sum of the angles of a triangle is 180 degrees

If all the angles must add to 180 and be the same….. Equiangular Triangle Triangle with 3 equal angles If all the angles must add to 180 and be the same….. Then, x+x+x = 180 3x = 180 X = 60

If you know 2 angles, then you can always figure out the 3rd Triangles If you know 2 angles, then you can always figure out the 3rd

Triangle Inequalities

Triangle Inequality Theorem: The sum of two smaller sides of a triangle must be greater than the length of the largest side. a + b > c a + c > b b + c > a Example: Determine if it is possible to draw a triangle with side measures 12, 11, and 17. 12 + 11 > 17  Yes 11 + 17 > 12  Yes 12 + 17 > 11  Yes Therefore a triangle can be drawn.

Angle Side Relationship The longest side is across from the largest angle. The shortest side is across from the smallest angle. 54 ° 37 89 B C A BC = 3.2 cm AB = 4.3 cm AC = 5.3 cm

Triangle Inequality – examples… For the triangle, list the angles in order from least to greatest measure. C A B 4 cm 6 cm 5 cm

Triangles

Congruent Triangles

The Idea of a Congruence Two geometric figures with exactly the same size and shape. A C B D E F

Congruent Triangles Have congruent corresponding parts, sides and angles May be flipped and/or rotated BE CAREFUL WHEN YOU NAME THE SHAPE

Congruent Triangles are named such that the corresponding angles are in the same order

Write the congruence statement B F E D A C

Name the congruent triangles List the congruent sides and angles Angles Sides