Angle-MANIA! A- sauce 7/13/2010

Do Now What the sum of the angles of any triangle?

Do Now What the sum of the angles of any triangle? 40 180 70 70

Do Now How do we draw a line that is 180 degrees? Take a couple seconds think about this. 180°

Objective Find the complements and supplements of angles Identify and Find the angles of parallel lines and transversal Find exterior angles of a triangle. All of this will be done by solving equations

Purpose Learn some geometric terms Develop your algebra and equations solving skills.

Relationships b/t Angles If we know the relationships we can set up equations to solve for the measurement of angles we do not know.

Relationships b/t Angles Supplementary Angles sum of their angles is 180° Example: 100° and 80° are supplementary. We say, 100 ° is the supplement of 80 ° 100° 80°

Relationships b/t Angles Similarly, Complementary Angles sum of their angles is 90° Example: 35° and 55° are complementary. We say, 35° is the complement of 55° 35° 55°

Finding the Supplement Find the supplement of an angle measured as 45. We subtract 45 number from 180 to find the supplement X + 45 = 180 X = 180 − 45 X= ? 135°

Finding the Complement Very similar to Finding the Supplement. 45 + X = 90 X = ? 45° is a Complement of itself.!!! 45 WHEWW!!

Find the measurement of angles Solve for x 7x + 3x = 180 10x = 180 10x/10 = 180/10 X= 18 Use x to find the measurements Without knowing these angles are supplementary would we have been able to find the measurement of these angles? 7x = 7 · 18 = 126 3x = 3 · 18 = 54 126 + 54 = 180

Pause. Complete Problems on Guided notes called Find the supplement and complement of angle measured at 88 degrees.   Given ∠A and ∠B are supplementary and m∠A = 7x + 4 & m∠B = 4x + 9. Find each angles measure.

Angles of Parallel Lines First let’s consider a parallelogram

Angles of Parallel Lines What’s this red line called? Transversal

Transversal Transversal is a line that intersects two or more lines that lie in the same plane in different points. A transversal of parallel lines creates Equal corresponding angles Equal alternate interior angles Supplementary interior angles on the same side of the transversal Equal vertical angles

A A Corresponding Angles Corresponding Angles are in the same position around both lines. A

Alternate Interior Angles We can easily see that the angles are inside (between) the parallel lines. Why alternate? A

Interior angles on the same side Also called consecutive interior angles Why are they supplementary? A A

A A Vertical Angles Vertical angles are opposite of one another. What is another pair of vertical angles?

Identify Relationship b/t angles Angle 2 and Angle 6 Consecutive Angles Angle 4 and Angle 5 Corresponding Angles Angle 1 and Angle 6 Alternate Interior Angles Angle 5 and Angle 8 Vertical Angles

Find the measurement of angles Angle 1 = 56°, because vertical angle Angle 2= 180- 56 = 124°, supplementary In partners, find the measurement of the rest of the angles and why. Angle 3 = 124 ° Angle 4 = 56° Angle 5 = 124° Angle 6 = 124° Angle 7 = 56°

Solve for variables x equals 108 because it’s the supplement of 72 AND? y = 36 because 3y and x are corresponding angles. 3y = 108 y = 36 (3z + 18) is equal to 108 degrees. Why? What is z?

Pause Practice Problems Find the value of x and y. (Hint: Extend lines to determine transversal) If < 2 = 35°, find the measure of the rest of the angles

Exterior Angles of Triangle An exterior angle of a triangle is equal to the sum of the opposite interior angles. SO w = x + y What other relations do we know?

Find the exterior angle First we notice this is an isosceles triangle. We also need to find measure of base angles So we can use this equation: 44 + 2y = 180 Why? Y

Find the exterior angle 44 + 2y = 180 (subtract 44 from both sides) 2y = 136 (divide both sides by 2) y = 68 So now we can find x because?

Find the exterior angle X = 44 + 68 = 112 because x is an exterior angle of the triangle .

Pause Practice Problem. Find x.

Proof of Triangle Theorem Sum of angles in triangle equal to 180°

Proof of Triangle Theorem Next we draw parallel to base ‘a’ through point P (intersection of ‘c’ and ‘b’) First, lets label the sides and angles A b c C B a