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Unit 3: Geometry Gallery KEY STANDARDS: Students will discover, prove, and apply properties of triangles, quadrilaterals, and other polygons. a. Determine the sum of interior and exterior angles in a polygon. b. Understand and use the triangle inequality, the side-angle inequality, and the exterior angle inequality. c. Understand and use congruence postulates and theorems for triangles (SSS, SAS, ASA, AAS, and HL). d. Understand, use, and prove properties of and relationships among special quadrilaterals: parallelogram, rectangle, rhombus, square, trapezoid, and kite. e. Find and use points of concurrency in triangles: incenter, orthocenter, circumcenter, and centroid. KEY IDEAS 1. In a regular polygon, all side lengths are congruent, and all angles are congruent. 2. The following information applies to interior and exterior angles of polygons: • The Interior Sum Theorem for triangles states that the sum of the measures of the three interior angles of a triangle always equals 180°. • The sum of the measures of the interior angles of a convex polygon is found by using the formula 180(n − 2), where n is the number of sides of the polygon. Examples: a) What is the sum of the interior angles of a convex octagon? Use 180(n-2) and plug in 8 for n, since an octagon has 8 sides. 180(n-2) = 180(8-2) = 180(6) = 1080; so the sum of the interior angles of a convex octagon is 1080 . b) Find the sum of the interior angles of a convex 23-sided figure. Use 180(n-2) and plug in 23 for n. 180(23-2) = 180(21) = 3780; so the sum of the interior angles of a convex 23-sided figure is 3780 . *The measure of each interior angle of a regular n-gon is found by using the formula 180(n 2) Example: Find each interior angle of a regular hexagon. n Use 180(n-2) to find the sum of the interior angles of a hexagon: 180(n-2) = 180(6-2) = 180(4) = 720. Now divide that total by the number of sides in the polygon: 720 divided by 6 = 120; so each interior angle of a regular hexagon = 120 . • The exterior angle of a polygon is an angle that forms a linear pair with one of the angles of the polygon. In this figure, <QRS is an exterior angle of triangle PQR. • Interior angles and their adjacent exterior angles are always supplementary. The sum of the degree measures of the two angles is 180 . . In this figure, m<QRS + m<QRP = 180 . • The remote interior angles of a triangle are the two angles nonadjacent to the exterior angle. In this figure, <P and <Q are the remote interior angles to exterior <QRS. (they don’t touch that angle at all!) • The measure of the exterior angle of a triangle equals the sum of the measures of the two remote interior angles. In this figure, m<QRS = m<P + m<Q. The Exterior Angle Inequality states that an exterior angle of a triangle is greater than either of the remote interior angles. In this figure, m<QRS is >m<Q and >m<P. • The Exterior Angle Sum Theorem states that if a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is 360°. The corollary that 360 follows states that the measure of each exterior angle of a regular n-gon is . n Examples: a) Find the sum of the interior angles of a pentagon: 360 of a decagon: 360 Of a figure with 29 sides: 360 360 360 b) Find each exterior angle of a pentagon: 72 , so each exterior angle of a pentagon n 5 is 72 . The following theorems apply to triangles: 1. Theorem: If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. 2. Theorem: If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. In BCA, the longest side is AB, so the largest angle is the angle opposite AB, which is <C. The shortest side is BC, so the smallest angle is the one opposite BC, which is <A. In DFE, the largest angle is <D, so the longest side is the side opposite <D, which is FE. The smallest angle is <E, so the shortest side is the side opposite <E, which is DF. 3. Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side (the Triangle Inequality Theorem). Do these three lengths determine a triangle? 2, 6, 3 no; because 2 + 3 is < 6. 6, 5, 8? Yes; because 6 + 5 is > 8. Congruent Triangles KEY IDEAS 1. The symbol ≅ means “is congruent to.” If ABC ≅ XYZ, then AB ≅ XY, BC ≅YZ, AC ≅ XZ, ∠A≅∠X , ∠B ≅∠Y, and ∠C ≅∠Z. The following key ideas are all theorems. 2. Theorem: If two triangles are congruent, then the corresponding parts of the two congruent triangles are congruent. 3. SSS Theorem: If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. 4. SAS Theorem: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. 5. ASA Theorem: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent. ASA leads to the AAS corollary. If two angles of a triangle are known, then the third angle is also known. 6. HL Theorem: If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the two triangles are congruent. 7. Other right triangle congruencies include: Hypotenuse-Angle (HA), Leg-Angle (LA) and Leg- Leg (LL). Remember: Mark your shared sides and vertical angles, and then label each triangle with the corresponding letters (such as ASA); if the triangle letters match, and it’s one of the postulates or theorems we’ve discussed, they are congruent by that postulate or theorem. Example: State the theorem that supports that the two triangles are congruent and write the congruence statement. Explain your reasoning. Solution: The theorem is SAS. The congruence statement is ABC ≅ DEF Reasoning: • The single tick marks on AB and DE indicate that the sides are congruent. • The two tick marks on AC and DF indicate that these sides are congruent. • The angle markings on ∠A and ∠D indicate that these two angles are congruent. The drawing shows two congruent sides and the angles between these two sides are congruent (SAS). POINTS OF CONCURRENCY IN TRIANGLES KEY IDEAS 1. Two or more lines that intersect in one point are concurrent lines. This intersection point is known as the point of concurrency. a. Centroid is the point of concurrency of the medians of a triangle. A median of a triangle is a segment that joins a vertex of a triangle to the midpoint of the opposite side. The point of concurrency of the medians (the point of intersection) is called the centroid of the triangle. The centroid of ABC is shown in this diagram. b. Circumcenter is the point of concurrency of the perpendicular bisectors of the sides of a triangle. This diagram shows that the three perpendicular bisectors of�� ABC are concurrent at a single point. This point of concurrency is called the circumcenter of the triangle. This point is also the center of the circle circumscribed about�� ABC. Notice that the circle passes through all three vertices of�� ABC. c. Incenter is the point of concurrency of the angle bisectors of a triangle. This diagram shows the angle bisectors of ABC. The angle bisectors intersect at a point of concurrency known as the incenter of the triangle. It is the center of the circle that can be inscribed in ABC. 5. Orthocenter is the point of concurrency of the altitudes of a triangle. An altitude of a triangle is a perpendicular segment from a vertex of the triangle to the line containing the opposite side. The point of concurrency of the lines that contain the altitudes of a triangle (the point of intersection) is called the orthocenter of the triangle. This diagram shows the orthocenter of ABC. REVIEW EXAMPLES 1) What is the degree measure of the exterior angle in this figure? 2) Consider CDE. List the sides in order by length from the greatest to the least. 3) In ECD, m∠E =136 , m∠C =17°, and m∠D= 27°. Which statement must be true? (DRAW THE TRIANGLE!) A. CD< DE B. DE <CD C. CE>CD D. DE>CE 4) Which set could be the lengths of the sides of a triangle? A. 15 cm, 18 cm, 26 cm B. 16 cm, 16 cm, 32 cm C. 17 cm, 20 cm, 40 cm D. 18 cm, 22 cm, 42 cm 5) Find the sum of the interior angles of a 22-sided figure. A. 3240 B. 3960 C. 3600 D. 360 6) Which set of relationships is sufficient to prove that the triangles in this figure are congruent? (MARK THE TRIANGLES!) A. PR ≅ SU, PQ ≅ ST,∠Q ≅∠U B. PQ ≅ PR, ST ≅ SU, RQ ≅TU C. RQ ≅TU,∠R ≅∠U,∠P ≅∠S D. ∠P ≅∠S,∠R ≅∠U,∠Q ≅∠T 7) A student wants to inscribe a circle inside of a triangle. Which of the following should the student construct to locate the incenter of the triangle? A. the medians of the triangles B. the altitudes of the triangles C. the angle bisectors of the triangle D. the perpendicular bisectors of the sides of the triangle 8) Jay constructed a line segment from each vertex that was perpendicular to the line containing the opposite side of a triangle. At what point of concurrency did the lines meet? A. the incenter B. the centroid C. the orthocenter D. the circumcenter 9) Find each exterior angle of a dodecagon (12-sided figure). 10) True or False: The diagonals of a rectangle are congruent.

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posted: | 10/29/2011 |

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