Pearson Unit 1 Topic 3: Parallel & Perpendicular Lines 3-5: Parallel Lines and Triangles Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

TEKS Focus: (6)(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems. (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. (1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Terms – Classifying by Angles Vocabulary Term Definition Diagram Three acute angles. Acute Equiangular Three equal angles. One 90⁰ angle. Right Obtuse One obtuse angle.

Terms – Classifying by Sides Vocabulary Term Definition Picture Equilateral Three congruent sides. At least two congruent sides. Isosceles Scalene No congruent sides.

An interior angle is formed by two sides of a triangle An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and extension of an adjacent side. 4 is an exterior angle. Exterior Interior 3 is an interior angle.

The remote interior angles of 4 are 1 and 2. Each exterior angle has two remote interior angles. The 2 remote interior angles of a triangle are the two nonadjacent interior angles corresponding to each exterior angle of the triangle. 4 is an exterior angle. The remote interior angles of 4 are 1 and 2. Exterior Interior 3 is an interior angle.

An auxiliary line is a line that you can add to a diagram to help explain relationships in proofs.

Investigation: Angle Relationships in a Triangle Sketchpad Straightedge and protractor

The Converse of the Corresponding Angles Postulate is used to construct parallel lines. The Parallel Postulate guarantees that for any line ℓ, you can always construct a parallel line through a point that is not on ℓ. Note

Example: 1 After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mXYZ. mXYZ + mYZX + mZXY = 180° Sum. Thm Substitute 40 for mYZX and 62 for mZXY. mXYZ + 40 + 62 = 180 mXYZ + 102 = 180 Simplify. mXYZ = 78° Subtract 102 from both sides.

Example: 2 Lin. Pair Thm. and  Add. Post. Substitute 62 for mYXZ. After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mYWZ. Step 1 Find mWXY. mYXZ + mWXY = 180° Lin. Pair Thm. and  Add. Post. 62 + mWXY = 180 Substitute 62 for mYXZ. mWXY = 118° Subtract 62 from both sides. Step 2 Find mYWZ. mYWX + mWXY + mXYW = 180° Sum. Thm Substitute 118 for mWXY and 12 for mXYW. mYWX + 118 + 12 = 180 mYWX + 130 = 180 Simplify. mYWX = 50° Subtract 130 from both sides. mYWX = m YWZ =50°

Example: 3 Use the diagram to find mMJK. mMJK + mJKM + mKMJ = 180° Sum. Thm Substitute 104 for mJKM and 44 for mKMJ. mMJK + 104 + 44= 180 mMJK + 148 = 180 Simplify. mMJK = 32° Subtract 148 from both sides.

Example: 4 One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle? Let the acute angles be A and B, with mA = 2x°. mA + mB = 90° Acute s of rt. are comp. 2x + mB = 90 Substitute 2x for mA. mB = (90 – 2x)° Subtract 2x from both sides.

Example: 5 Find mB. mB = 2x + 3 = 2(26) + 3 = 55° Ext.  Thm. 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°

Example: 6 Find mACD. mACD = 6z – 9 = 6(25) – 9 = 141° Ext.  Thm. mACD = mA + mB Ext.  Thm. Substitute 6z – 9 for mACD, 2z + 1 for mA, and 90 for mB. 6z – 9 = 2z + 1 + 90 6z – 9 = 2z + 91 Simplify. Subtract 2z and add 9 to both sides. 4z = 100 z = 25 Divide by 4. mACD = 6z – 9 = 6(25) – 9 = 141°

EXTRA EXAMPLES NOT USED IN COMPOSITION BOOK FOLLOW. ALSO REMEMBER TO LOG-ON TO YOUR PEARSON ACCOUNT TO LOOK AT OTHER EXAMPLES BEFORE BEGINNING THE ON-LINE HW AND THE WRITTEN HW.

Example: 7 The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle? Let the acute angles be A and B, with mA = 63.7°. mA + mB = 90° Acute s of rt. are comp. 63.7 + mB = 90 Substitute 63.7 for mA. mB = 26.3° Subtract 63.7 from both sides.

Example: 8 The measure of one of the acute angles in a right triangle is x°. What is the measure of the other acute angle? Let the acute angles be A and B, with mA = x°. mA + mB = 90° Acute s of rt. are comp. x + mB = 90 Substitute x for mA. mB = (90 – x)° Subtract x from both sides.

Example: 9 The measure of one of the acute angles in a right triangle is 48 . What is the measure of the other acute angle? 2° 5 Let the acute angles be A and B, with mA = 48 . 2° 5 mA + mB = 90° Acute s of rt. are comp. 48 + mB = 90 2 5 Substitute 48 for mA. 2 5 mB = 41 3° 5 Subtract 48 from both sides. 2 5