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Name: ___________________________________________________________________________ Module 1, Topic G, Lesson 33 & 34 β Module 1 Review Fact/Property Two angles that form a linear pair are supplementary. Guiding Questions/Applications Notes/Solutions 133 o b The sum the measures of all adjacent angles formed by three or more rays with the same vertex is 360Λ. Vertical angles have equal measure. Name the pairs of vertical angles that are equal in measure. The bisector of an angle is a ray in the interior of the angle such that the two adjacent angles formed by it have equal measure. In the diagram below, π΄πΆ is the bisector of β π΄π΅π·, which measures 64°. What is the measure of β π΄π΅πΆ? Date: ________________ The perpendicular bisector of a segment is the line that passes through the midpoint of a line segment and is perpendicular to the line segment. In the diagram below, π·πΆ is the β₯ bisector of π΄π΅, and πΆπΈ is the β bisector of β π΄πΆπ·. Find the measures of π΄πΆ and β πΈπΆπ·. The sum of the 3 angle measures of any triangle is 180Λ. Given the labeled figure below, find the measures of β π·πΈπ΅ and β π΄πΆπΈ. When one angle of a triangle is a right angle, the sum of the measures of the other two angles is 90Λ. This fact follows directly from the preceding one. How is simple arithmetic used to extend the angle sum of a triangle property to justify this property? The sum of the measures of two angles of a triangle equals the measure of the opposite exterior angle. Use the exterior angle theorem to find the m<ACE. Base angles of an isosceles triangle are congruent. The triangle in the figure above is isosceles. How do we know this? All angles in an equilateral triangle have equal measure. If the figure above is changed slightly, it can be used to demonstrate the equilateral β³ property. Explain how this can be demonstrated. If a transversal intersects two Why does the property specify parallel lines? parallel lines, then the measures of the corresponding angles are equal. If a transversal intersects two lines such that the measures of the corresponding angles are equal, then the lines are parallel. Find the measure of the corresponding angle. If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary. Use the diagram below (π΄π΅ || πΆπ·) to show that β π΄πΊπ» and β πΆπ»πΊ are supplementary. If a transversal intersects two lines such that the same side interior angles are supplementary, then the lines are parallel. Use the diagram below (π΄π΅ || πΆπ·) to show that β π΄πΊπ» and β πΆπ»πΊ are supplementary. 1. If a transversal intersects two parallel lines, then the measures of alternate interior angles are equal. 2. Name both pairs of alternate interior angles in the diagram above. How many different angle measures are in the diagram? If a transversal intersects two lines such that measures of the alternate interior angles are equal, then the lines are parallel. Although not specifically stated here, the property also applies to alternate exterior angles. Why is this true? Given two triangles β³ π΄π΅πΆ and β³ π΄βπ΅βπΆβ so that π΄π΅ = π΄β²π΅β² (Side), πβ π΄ = πβ π΄β² (Angle), π΄πΆ = π΄β²πΆβ²(Side), then the triangles are congruent. [SAS] The figure below is a parallelogram ABCD. What parts of the parallelogram satisfy the SAS triangle congruence criteria for β³ π΄π΅π· and β³ πΆπ·π΅? Given two triangles β³ π΄π΅πΆ and β³ π΄βπ΅βπΆβ, if πβ π΄ = πβ π΄β² (Angle), π΄π΅ = π΄β²π΅β² (Side), and πβ π΅ = πβ π΅β² (Angle), then the triangles are congruent. [ASA] In the figure below, β³ πΆπ·πΈ is the image of reflection of β³ π΄π΅πΈ across line πΉπΊ. Which parts of the triangle can be used to satisfy the ASA congruence criteria? Given two triangles β³ π΄π΅πΆ and β³ π΄βπ΅βπΆβ, if π΄π΅ = π΄β²π΅β² (Side), π΄πΆ = π΄β²πΆβ² (Side), and π΅πΆ = π΅β²πΆβ² (Side), then the triangles are congruent. [SSS] β³ π΄π΅πΆ andβ³ π΄π·πΆ are formed from the intersections and center points of circles π΄ and πΆ. Prove β³ π΄π΅πΆ β β³ π΄πΆπ· by SSS. Given two triangles β³ π΄π΅πΆ and β³ π΄βπ΅βπΆβ, if π΄π΅ = π΄β²π΅β² (Side), πβ π΅ = πβ π΅β² (Angle), and πβ πΆ = πβ πΆβ² (Angle), then the triangles are congruent. The SAA congruence criterion is essentially the same as the ASA criterion for proving triangles congruent. Why is this true? A [AAS] B E D C Given two right triangles π΄π΅πΆ and π΄β²π΅β²πΆβ² with right angles β π΅ and β π΅β², if π΄π΅ = π΄β²π΅β² (Leg) and π΄πΆ = π΄β²πΆβ² (Hypotenuse), then the triangles are congruent. In the figure below, πΆπ· is the perpendicular bisector of π΄π΅ and β³ABC is isosceles. Name the two congruent triangles appropriately and describe the necessary steps for proving them congruent using HL. [HL] The opposite sides of a parallelogram are parallel and congruent. The opposite angles of a parallelogram are congruent. The diagonals of a parallelogram bisect each other. The mid-βsegment of a triangle is a line segment that connects the midpoints of two sides of a triangle; the mid-βsegment is parallel to the third side of the triangle and is half the length of the third side. π·πΈ is the mid-βsegment of β³ π΄π΅πΆ. Find the perimeter of β³ π΄π΅πΆ, given the labeled segment lengths. The three medians of a triangle are concurrent at the centroid; the centroid divides each median into two parts, from vertex to centroid and centroid to midpoint in a ratio of 2:1. If π΄πΈ, π΅πΉ, and πΆπ· are medians of β³ π΄π΅πΆ, find the lengths of segments π΅πΊ, πΊπΈ, and πΆπΊ, given the labeled lengths. Review Questions: Use any of the assumptions, facts, and/or properties presented in the tables above to find x and y in each figure below. C D 1. π₯ = 52 o 108o A F E π¦ = A C 2. D 108o 52 H o F E You will need to draw an auxiliary line to solve this problem. A 109o JE B y 107 x G I x π₯ = A B o y x y x C 64 o D B 68 o L π¦ = H F J o x y K M 3. Given the labeled diagram at the right, prove that β πππ β β πππ. Y Find the perimeter of parallelogram π΄π΅πΆπ·. Justify your solution. π΄πΆ = 34 π΄π΅ = 26 π΅π· = 28 Find the perimeter of β³ πΆπΈπ·. Justify your solution. 6. ππ = 12 ππ = 20 ππ = 24 πΉ, πΊ , and π» are midpoints of the sides on which they are located. Find the perimeter of β³ πΉπΊπ». Justify your solution. 7. Z W 5. V X 4. X D πΆ is the centroid of β³ π ππ. π πΆ = 16, πΆπΏ = 10, ππ½ = 21 ππΆ = ππΆ = πΎπΆ =