PARALLEL LINES CUT BY A TRANSVERSAL

DEFINITIONS PARALLEL TRANSVERSAL ANGLE VERTICAL ANGLE CORRESPONDING ANGLE ALTERNATE INTERIOR ANGLE ALTERNATE EXTERIOR ANGLE

DEFINITIONS SUPPLEMENTARY ANGLE COMPLEMENTARY ANGLE CONGRUENT

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 1 and < 2 are called SUPPLEMENTARY ANGLES DEFINITION:They form a straight angle measuring 180 degrees.

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 2 and < 3 < 3 and < 4 < 4 and < 1 < 5 and < 6 Name other supplementary pairs: < 6 and < 7 < 7 and < 8 < 8 and < 5

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 1 and < 3 are called VERTICAL ANGLES They are congruent m<1 = m<3 DEFINITION: The angles formed from two lines are crossing.

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 2 and < 4 < 6 and < 8 Name other vertical pairs: < 5 and < 7

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 1 and < 5 are called CORRESPONDING ANGLES They are congruent m<1 = m<5 DEFINITION: Corresponding angles occupy the same position on the top and bottom parallel lines.

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 2 and < 6 < 3 and < 7 Name other corresponding pairs: < 4 and < 8

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 4 and < 6 are called ALTERNATE INTERIOR ANGLES They are congruent m<4 = m<6 DEFINITION:Alternate Interior on the inside of the two parallel lines and on opposite sides of the transversal.

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 3 and < 5 Name other alternate interior pairs:

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 1 and < 7 are called ALTERNATE EXTERIOR ANGLES They are congruent m<1 = m<7 Alternate Exterior on the outside of the two parallel lines and on opposite sides of the transversal.

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 2 and < 8 < 1 and < 7 Name other alternate exterior pairs:

Parallel lines cut by a transversal 2 1 3 4 6 5 7 8 < 4 and < 5 are called CONSECUTIVE INTERIOR ANGLES The sum is 180. m<4 = m<5 DEFINITION: Consecutive Interior on the inside of the two parallel lines and on same side of the transversal. Sum = 180

TRY IT OUT 2 1 3 4 6 5 7 8 120 degrees The m < 1 is 60 degrees. What is the m<2 ? 120 degrees

TRY IT OUT 2 1 3 4 6 5 7 8 60 degrees The m < 1 is 60 degrees. What is the m<5 ? 60 degrees

TRY IT OUT 2 1 3 4 6 5 7 8 60 degrees The m < 1 is 60 degrees. What is the m<3 ? 60 degrees

TRY IT OUT 120 60 60 120 120 60 60 120

TRY IT OUT 2x + 20 x + 10 What do you know about the angles? Write the equation. Solve for x. SUPPLEMENTARY 2x + 20 + x + 10 = 180 3x + 30 = 180 3x = 150 x = 50

TRY IT OUT 3x - 120 2x - 60 What do you know about the angles? Write the equation. Solve for x. ALTERNATE INTERIOR 3x - 120 = 2x - 60 x = 60 Subtract 2x from both sides Add 120 to both sides

Warm Up Identify each angle pair. 1. 1 and 3 2. 3 and 6 3. 4 and 5 4. 6 and 7 corr. s alt. int. s alt. ext. s same-side int s

Example 1: Using the Corresponding Angles Postulate Find each angle measure. A. mECF x = 70 Corr. s Post. mECF = 70° B. mDCE 5x = 4x + 22 Corr. s Post. x = 22 Subtract 4x from both sides. mDCE = 5x = 5(22) Substitute 22 for x. = 110°

Check It Out! Example 1 Find mQRS. x = 118 Corr. s Post. mQRS + x = 180° Def. of Linear Pair mQRS = 180° – x Subtract x from both sides. = 180° – 118° Substitute 118° for x. = 62°

WEBSITES FOR PRACTICE Ask Dr. Math: Corresponding /Alternate Angles Project Interactive: Parallel Lines cut by Transversal

Triangle Sum Theorem The sum of the angle measures in a triangle equal 180° 3 2 1 m<1 + m<2 + m<3 = 180°

Corollary A corollary to a theorem is a statement that follows directly from that theorem

TRIANGLE ANGLE SUM THEOREM COROLLARIES If 2 angles of 1 triangle are congruent to 2 angles of another triangle, then the 3rd angles are congruent The acute angles of a right triangle are complementary The measure of each angle of an equiangular triangle is 60o A triangle can have at most 1 right or 1 obtuse angle

Exterior Angle Theorem (your new best friend) The measure of an exterior angle in a triangle is the sum of the measures of the 2 remote interior angles exterior angle remote interior angles 3 2 1 4 m<4 = m<1 + m<2

REMOTE INTERIOR ANGLE In any polygon, a remote interior angle is an interior angle that is not adjacent to a given exterior angle A and B are remote to angle 1

Exterior Angle Theorem interior In the triangle below, recall that 1, 2, and 3 are _______ angles of ΔPQR. Angle 4 is called an _______ angle of ΔPQR. exterior An exterior angle of a triangle is an angle that forms a _________ with one of the angles of the triangle. linear pair In ΔPQR, 4 is an exterior angle at R because it forms a linear pair with 3. ____________________ of a triangle are the two angles that do not form a linear pair with the exterior angle. Remote interior angles In ΔPQR, 1, and 2 are the remote interior angles with respect to 4. 1 2 3 4 P Q R

Exterior Angle Theorem In the figure below, 2 and 3 are remote interior angles with respect to what angle? 5 1 2 3 4 5

an example with numbers find x & y 82° x = 68° y = 112° 30° x y Determine the measure of <4, If <3 = 50, <2 = 70 40x 10x2 30x find all the angle measures Do you hear the sirens????? 80°, 60°, 40°

Exterior Angle Theorem

Example 3: Applying the Exterior Angle Theorem Find mB. mA + mB = mBCD Ext.  Thm. Substitute 15 for mA, 2x + 3 for mB, and 5x – 60 for mBCD. 15 + 2x + 3 = 5x – 60 2x + 18 = 5x – 60 Simplify. Subtract 2x and add 60 to both sides. 78 = 3x 26 = x Divide by 3. mB = 2x + 3 = 2(26) + 3 = 55°

There are several ways to prove certain triangles are similar There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent.

Example 1: Using the AA Similarity Postulate Explain why the triangles are similar and write a similarity statement. Since , B  E by the Alternate Interior Angles Theorem. Also, A  D by the Right Angle Congruence Theorem. Therefore ∆ABC ~ ∆DEC by AA~.

Check It Out! Example 1 Explain why the triangles are similar and write a similarity statement. By the Triangle Sum Theorem, mC = 47°, so C  F. B  E by the Right Angle Congruence Theorem. Therefore, ∆ABC ~ ∆DEF by AA ~.