Angles, Triangles, Transformations and Proofs Packet 1 Notes and some practice are included Homework will be assigned on a daily basis Topics Covered: Vocabulary Angle relationships Parallel Lines & Transversals Triangle Theorems Transformations Triangle Congruence postulates Proofs Test for the unit is 8/21 Name:
Vocabulary Adjacent Angles: Angles in the same plane that have a common vertex and a common side, but no common interior points. Alternate Exterior Angles: Alternate exterior angles are pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on opposite sides of the transversal and are outside the other two lines. When the two other lines are parallel, the alternate exterior angles are equal. Alternate Interior Angles: Alternate interior angles are pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on opposite sides of the transversal and are in between the other two lines. When the two other lines are parallel, the alternate interior angles are equal. Angle: Angles are created by two distinct rays that share a common endpoint (also known as a vertex). ABC or B denote angles with vertex B. Bisector: A bisector divides a segment or angle into two equal parts. Centroid: The point of concurrency of the medians of a triangle. Circumcenter: The point of concurrency of the perpendicular bisectors of the sides of a triangle. Coincidental: Two equivalent linear equations overlap when graphed. Complementary Angles: Two angles whose sum is 90 degrees. Congruent: Having the same size, shape and measure. Two figures are congruent if all of their corresponding measures are equal. Congruent Figures: Figures that have the same size and shape. Corresponding Angles: Angles that have the same relative positions in geometric figures. Corresponding Sides: Sides that have the same relative positions in geometric figures Dilation: Transformation that changes the size of a figure, but not the shape. Endpoints: The points at an end of a line segment Equiangular: The property of a polygon whose angles are all congruent. Equilateral: The property of a polygon whose sides are all congruent. Exterior Angle of a Polygon: an angle that forms a linear pair with one of the angles of the polygon. Incenter: The point of concurrency of the bisectors of the angles of a triangle. Intersecting Lines: Two lines in a plane that cross each other. Unless two lines are coincidental, parallel, or skew, they will intersect at one point. Intersection: The point at which two or more lines intersect or cross. Line: One of the basic undefined terms of geometry. Traditionally thought of as a set of points that has no thickness but its length goes on forever in two opposite directions. AB denotes a line that passes through point A and B. Line Segment or Segment: The part of a line between two points on the line. AB denotes a line segment between the points A and B. Linear Pair: Adjacent, supplementary angles. Excluding their common side, a linear pair forms a straight line. 180 ( n 2) Measure of each Interior Angle of a Regular n-gon: n Orthocenter: The point of concurrency of the altitudes of a triangle. Parallel Lines: Two lines are parallel if they lie in the same plane and they do not intersect. Perpendicular Lines: Two lines are perpendicular if they intersect at a right angle. Plane: One of the basic undefined terms of geometry. Traditionally thought of as going on forever in all directions (in two-dimensions) and is flat (i.e., it has no thickness). Point: One of the basic undefined terms of geometry. Traditionally thought of as having no length, width, or thickness, and often a dot is used to represent it. Proportion: An equation which states that two ratios are equal.
Ratio: Comparison of two quantities by division and may be written as r/s, r:s, or r to s. Ray: A ray begins at a point and goes on forever in one direction. Reflection: A transformation that "flips" a figure over a line of reflection Reflection Line: A line that is the perpendicular bisector of the segment with endpoints at a pre-image point and the image of that point after a reflection. Regular Polygon: A polygon that is both equilateral and equiangular. Remote Interior Angles of a Triangle: the two angles non-adjacent to the exterior angle. Rotation: A transformation that turns a figure about a fixed point through a given angle and a given direction. Same-Side Interior Angles: Pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on the same side of the transversal and are between the other two lines. When the two other lines are parallel, same-side interior angles are supplementary. Same-Side Exterior Angles: Pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on the same side of the transversal and are outside the other two lines. When the two other lines are parallel, same-side exterior angles are supplementary. Scale Factor: The ratio of any two corresponding lengths of the sides of two similar figures. Similar Figures: Figures that have the same shape but not necessarily the same size. Skew Lines: Two lines that do not lie in the same plane (therefore, they cannot be parallel or intersect). Sum of the Measures of the Interior Angles of a Convex Polygon: 180º(n 2). Supplementary Angles: Two angles whose sum is 180 degrees. Transformation: The mapping, or movement, of all the points of a figure in a plane according to a common operation. Translation: A transformation that "slides" each point of a figure the same distance in the same direction Transversal: A line that crosses two or more lines. Vertical Angles: Two nonadjacent angles formed by intersecting lines or segments. Also called opposite angles.
Name: Date: Complementary, Supplementary, Linear Pairs & Vertical Angles Geometry Symbols: Angle Degree Right Angle Perpendicular Segment AB Ray CD Line EF Measure Naming Angles: Name this angle 4 different ways: Naming Segments What is the name of the top side: Angle Bisector Cuts an angle into TWO congruent angles. Example: Solve for x: Linear Pair Two angles that are side-by-side, share a common vertex, share a common ray, & create a straight line. EQUATION: Examples: Solve for x: o o
Supplementary Angles Two angles that add up to. EQUATION: Examples: Solve for x: o 105 o 2x+5 3x+11 119-x o o x and y are supplementary angles. m x = 47. Find m y. o One of two supplementary angles is 46 degrees more than its supplement. Find the measure of both angles. Complementary Angles Two angles that add up to. EQUATION: Examples: Solve for x: o o x 70 o o One of two complementary angles is 16 degrees less than its complement. Find the measure of both angles. 2x x+30 Vertical Angles 2 angles that share a common vertex & their sides form two pairs of opposite rays. EQUATION: Examples: Solve for x: 2x-17 122 o o o x -12 x 130 x+135 z y
Name: Date: Complementary and Supplementary Angles Notes TYPES OF ANGLES: Sketch: 1. Acute: Acute angles have measures between 0 and 90. 2. Right: A right angle has measure equal to 90. 3. Obtuse: Obtuse angles have measures between 90 and 180. SPECIAL PAIRS OF ANGLES: 1. Complementary Angles: Pair of angles whose sum of measures equals 90. 40 and 50 angles are complementary angles because 40 + 50 = 90. Example: A 40 angle is called the complement of the 50 angle. Similarly, the 50 angle is the complement of the 40 angle. Practice: Find the complement of each angle. a) 35 b) 48 c) 12 2. Supplementary Angle: Pair of angles whose sum of measures equals 180. 60 and 120 angles are complementary angles because 60 + 120 = 180. Example: A 60 angle is called the supplement of the 120 angle. Similarly, the 120 angle is the supplement of the 60 angle. Practice: Find the supplement of each angle. a) 40 b) 126 c) 72 Can you think of a way to remember the difference between complementary and supplementary angles? 3. Angle Bisector: A ray (or line or segment) that divides an angle into two congruent angles (two angles with equal measure).
Alternate Interior Angles Opposite sides of the transversal & inside the parallels Are congruent Equation: angle = angle Consecutive Interior Angles Same side of the transversal & inside the parallels Are supplementary Equation: angle + angle = 180
Notes Isosceles and Equilateral Triangles Theorem Examples Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the sides are congruent. If RT RS, then /T > /S. Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. You can use these theorems to find angle measures in isosceles triangles. Find m E in DEF. m D m E 5x8 (3x 14)8 2x 14 Isosc. Thm. Substitute the given values. Subtract 3x from both sides. x 7 Divide both sides by 2. Thus m E 3(7) 14 358. If /N > /M, then LN LM. Find each angle measure. 1. m C 2. m Q 3. m H 4. m M
Notes Isosceles and Equilateral Triangles continued Equilateral Triangle Corollary If a triangle is equilateral, then it is equiangular. (equilateral equiangular ) Equiangular Triangle Corollary If a triangle is equiangular, then it is equilateral. (equiangular equilateral ) If /A > /B > /C, then AB BC CA. You can use these theorems to find values in equilateral triangles. Find x in STV. STV is equiangular. (7x 4)8 60 7x 56 Equilateral equiangular The measure of each of an equiangular is 60. Subtract 4 from both sides. x 8 Divide both sides by 7. Find each value. 5. n 6. x 7. VT 8. MN
TRANSFORMATIONS TRANSLATION SLIDE Translate right (x + #, y) Translate left (x #, y) Translate up (x, y + #) Translate down (x, y #) REFLECTION FLIP Across x-axis ( x, -y ) Change the sign of y Across y-axis ( -x, y ) Change the sign of x Across y = x ( y, x ) Swap both Across y = -x ( -y, -x ) Swap & Negate ROTATION TURN 90 CW & 270 CCW ( y, -x ) Negate x & swap 90 CCW & 270 CW ( -y, x ) Negate y & swap 180 either way ( -x, -y ) Negate both
Proving Triangles Congruent Angle-Side-Angle (ASA) postulate Angle-Angle-Side (AAS) theorem Side-Side-Side (SSS) post Side-Angle-Side post. (SAS)
Triangle Congruence Postulates Today s Question: What does it mean for two triangles to be congruent? (MCC9-12.G.SRT5, MCC9-12.G.CO.7-8) Congruent Triangles Side Side Side (SSS) Congruence Postulate three sides of one triangle are congruent to three sides of a second triangle Side Angle Side (SAS) Congruence Postulate two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle Angle Side Angle (ASA) Congruence Postulate two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle Angle Angle Side (AAS) Congruence Postulate two angles and a non-included side of one triangle are congruent to two angles and a non-included side of a second triangle Hypotenuse Leg (HL) Congruence Postulate In a right triangle, the hypotenuse and one leg is congruent to the hypotenuse and leg of another right triangle
Practice In each problem, determine if each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If they are, complete the congruence statement too. If none of these methods work based on the information given, write none. If congruent, finish the congruence statement. 1. BIG 2. SML 3. OPN 4. FLP 5. HOT 6. CLD 7. CAT 8. HIP 9. PAT
Euclidean Geometry Name Triangle Proof Tips General Directions: 1. You need 3 congruencies to prove that 2 triangles are congruent. 2. You cannot make up your own "given" information. 3. Every time you get a side or angle congruence, move to the next piece of given information. 4. When you run out of given information, try vertical angles or reflexive property. If you are given a midpoint of a segment, look for 2 congruent segments. If you are given an angle bisector, look for two congruent angles. If you are given parallel lines, look for Alternate Interior angles to be congruent. If you are given for perpendicular lines look for right angles. You can use the Reflexive Property and Vertical Angles without given information.
Name: Date: Matching: Use the choices listed at the bottom in the box for problems #1 4 Problem 1: Statement 1. LM LO 1. Given 2. MN ON 2. Given 3. LN LN 3. 4. LMN LON 4. Reason Problem 2: Statement 1. QS RT 1. Given 2. <R and <S are 2. Given right <s 3. R S 3. 4. 1 2 4. 5. QT QT 5. 6. QST TRQ 6. Reason Problem 3: Problem 4: Statement 1. GI KI 1. Given 2. HI JI 2. Given 3. GIH KIJ 3. 4. GIH KIJ 4. Statement 1. AC BD, AB CD 1. Given 2. 1 4, 2 3 2. 3. AD AD 3. 4. ADC DAB 4. Reason Reason Choices for problems #1 4 (some will be used more than once): AAS ASA Alternate Interior Angles are Given Reflexive Property SAS SSS Vertical Angles are
Fill in the blank proofs: Problem 5: Statement 1. I K 1. Given 2. IHJ KJH 2. Given 3. HJ HJ 3. 4. HJK JHI 4. Reason Problem 6: Statement 1. MLN ONL 1. Given 2. OLN 2. Given 3. 3. Reflexive Property 4. LNO NLM 4. Reason Problem 7: Statement 1. PQ QS 1. Given 2. TQ QP 2. Given 3. PQT RQS 3. 4. PQT SQR 4. Reason Problem 8: Statement Reason 1. UV UX 1. Given 2. <UWV and <UWX 2. Given are right <s 3. 3. Right Angle Congruence 4. 4. Reflexive Property 5. V X 5. Given 6. UWV UWX 6. Problem 9: Statement Reason 1. Y C 1. Given 2. YA AC 2. Given 3. 3. Vertical Angles are congruent 4. YZA CBA 4.
Problem 10: Statement 1. BAC DCA 1. Given 2. AB DC 2. Given 3. 3. 4. ABC CDA 4. Reason Problem 11: Statement 1. F I 1. Given 2. E H 2. Given 3. EG HJ 3. 4. EFG HIJ 4. Reason Problem 12: Statement 1. O M 1. Given 2. OL LM 2. Given 3. 3. 4. KLO NLM 4. 5. K N 5. CPCTC Reason Problem 13: Statement 1. P R 1. Given 2. PSQ RQS 2. Given 3. 3. Reflexive 4. PQS RSQ 4. Reason Problem 14: Statement Reason 1. AC BD 1. Given 2. B C 2. Given 3. CAD BDA 3. 4. 4. Reflexive Property 5. ACD 5.