Unit 2: Triangles and Quadrilaterals Lesson 2.1 pply Triangle Sum Properties Lesson 4.1 from textbook Objectives Classify angles by their sides as equilateral, isosceles, or scalene. Classify triangles by angle measure as acute, obtuse, right, or equiangular. Use the Triangle Sum Theorem and Exterior ngle Theorem to find angles measures in triangles. Classifying Triangles by Sides Classifying Triangles by ngles Example 1 Classify the triangle by its sides and angles. Interior ngles Exterior ngles
Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is. Example 2 Find the value of x. Then classify the triangle by its sides. x = Exterior ngle Theorem The measure of an exterior is equal to the sum of the measures of the two nonadjacent interior angles. Example 3 Find the value of x. x = Corollary to the Triangle Sum Theorem The acute angles of a right triangle are. Example 4 Find the values of x and y. x = y =
Unit 2: Triangles and Quadrilaterals Lesson 2.2 Use Isosceles and Equilateral Triangles Lesson 4.7 from Textbook Objectives Use properties of isosceles and equilateral triangles to find the measure of given angles. Use the Base ngles Theorem and Converse of the Base ngles Theorem to prove that parts of triangles are congruent. Isosceles Triangle Equilateral Triangle Base ngles Theorem If B C, then. Converse of the Base ngles Theorem If B C, then. Example 1 In DEF, DE DF. Name two congruent angles. Base ngle Theorem Corollaries Corollary #1 If a triangle is equilateral, then it is. Corollary #2 If a triangle is equiangular, then it is.
Example 2 Find the unknown measure. NO = m<u = Example 3 Find the values of x and y in the diagram. x = y = x = y = Example 4 Find the perimeter of the triangle. P =
Unit 2: Triangles and Quadrilaterals Lesson 2.3 Midsegment Theorem Lesson 5.1 from Textbook Objectives Identify and construct the midsegment of a triangle. Use the Midsegment Theorem to find the length of sides of the triangle and the length of the midsegment. Vocabulary Midsegment of a triangle B Construct the midsegment using sides B and BC and label it MP. Find the length of C. Find the length of MP. C What is the relationship between the midsegment and its parallel base? Midsegment Theorem The midsegment connecting two sides of a triangle is parallel to the third side and is. Example 1 UV and VW are midsegments of RST. RT = 16 m and VW = 6 m. Find UV and RS. UV = RS =
Example 2 In XYZ, XJ JY, YL LZ, and XK KZ. Complete the statement. JK JL XY YJ JL YL Example 3 Use GHJ, where, B, and C are midpoints of the sides. ) B = 3x + 8 and GJ = 2x + 24, what is B? B) GH = 7z 1 and BC = 4z 3, what is GH? Example 4 Place the figure in the coordinate plane so that it is convenient to measure the lengths of the sides of the triangle. Give the coordinates of the vertices. rectangle with a base length of 7 and a height of 6.
Unit 2: Triangles and Quadrilaterals Lesson 2.4: Polygons and Their ngles Measures Lesson 8.1 from textbook Objectives Use the Interior ngles of a Quadrilateral Theorem to find the measures of interior angles of a quadrilateral. Use the Polygon Interior ngles Theorem and the Polygon Exterior ngles Theorem to find the measures of interior and exterior angles of polygons Use properties of interior and exterior angles of polygons to solve for unknown angles in polygons. Recall: Names of Polygons Polygons are names by their number of sides. Number of Sides Type of Polygon Number of Sides 3 8 4 9 5 10 Type of Polygon 6 12 7 n INTERIOR NGLES IN POLYGONS In a polygon, two vertices that are endpoints of the same sides are called. of a polygon is a segment that joins two non-consecutive vertices. The diagonals from one vertex form triangles. The sum of the angles diagonals in a triangle is. The sum of the angles in the pentagon is. Polygon Interior ngles Theorem The sum of the measures of the interior angles of a convex n-gon is (n 2) 180 o. Example 1 Find the sum of the measures of the interior angles of a convex octagon.
Example2 The sum of the measures of the interior angles of a convex polygon is 900 o. Classify the polygon by the number of its sides. Example 3 Find the value of x in the diagram shown. 108 o 121 o x o 59 o Polygon Exterior ngles Theorem The sum of the measures of the exterior angles of a convex polygon, One at each vertex, is. m<1 + m<2 + m<3 + m<4 =. 1 2 5 3 4 *The sum of the measures of an interior angle and an exterior angle of a polygon is. m<4 + m<5 = Example 4 Find the value of x in the diagram. 2x o x o 89 o 67 o Example 5 Find the value of the exterior angles. regular pentagon. n isosceles triangle with angle measures 50 o, 50 o, and 80 o.
Unit 2: Triangles and Quadrilaterals Lesson 2.5: Use Properties of Parallelograms Lesson 8.2 from textbook Objectives Use the properties of a parallelogram to find the values of angles and side lengths of a parallelogram. Use properties of parallelograms to prove segment and angle congruence along with definitions, theorems, and postulates in the form of a proof. Vocabulary Parallelogram Q R parallelogram PQRS also named P CTIVITY: S lso,. 1. Use a ruler and find the lengths of all of the sides of parallelogram BCD. Write the measures on the figure. What do you notice about the measures of the opposite sides? B C 2. Use a protractor and find the measures of all of the angles. Write the measures on the figure. What do you notice about the measures of the opposite and consecutive angles? D 3. Use a ruler and draw the two diagonals of the parallelogram. What do you notice about the diagonals? Parallelogram Theorem #1 Parallelogram Theorem #2 If a quadrilateral is a parallelogram, then its opposite angles are congruent. Q R Q If a quadrilateral is a parallelogram, then its opposite sides are congruent. R P S P S
Example 1 BCD is a parallelogram. Find the value of x and y. x = y = 18 2x o B D 50 o y + 3 C Parallelogram Theorem #3 Parallelogram Theorem #4 If a quadrilateral is a parallelogram, then its If a quadrilateral is a parallelogram, then its consecutive angles are. diagonals. Q R Q R P S P S Example 2 Find the measure of the angles in the parallelogram. G H m<g = m<h = F 58 o I m<i = Example 3 Find the value of each variable in the parallelogram. m = n = L M 9 n 16 2n 4m O N Example 4 Find the indicated measure in parallelogram HIJK. HI = KH = IJ = HJ = m<kjh = m< JIH = J H 124 o 7 G 8 10 28 o 16 K I m<hji = m<jki =
Unit 2: Triangles and Quadrilaterals Lesson 2.6 Properties of Rhombuses, Rectangles, and Squares Lesson 8.4 from textbook Objectives Use the properties of rhombuses, rectangles, and squares to classify a quadrilateral. Use the properties of rhombuses, rectangles, and squares to find measures of angles and sides of parallelograms. Vocabulary Rhombus Rectangle Square Rhombus Theorem #1 Rectangle Theorem #1 Square Theorem #1 quadrilateral is a rhombus quadrilateral is a rectangle quadrilateral is a square if and only if it has four if and only if it has four if and only if it is a rhombus congruent sides. right angles. and a rectangle. Example 1 For any rhombus BCD, decide whether the For any rectangle MNOP, decide whether statement B C is true. Draw a diagram the statement OP MP is true. Draw a and explain your reasoning. diagram and explain your reasoning. Example 2 Classify the quadrilateral and explain your reasoning.
Rhombus Theorem #2 Rhombus Theorem #3 Rectangle Theorem #2 parallelogram is a rhombus parallelogram is a rhombus parallelogram is rectangle if and only if its diagonals are if and only if each diagonal if and only if its diagonals are perpendicular. bisects a pair of opposite angles. are congruent. ^ ^ > Example 3 > ^ Classify the quadrilateral. Explain your reasoning. Find the value of x and y. ^ ^ Classification > > ^ x = y = Example 4 The diagonals of rhombus WXYZ intersect at V. Given that m<xzy = 34 o and WV = 7, find the indicated measure. m<wzv = WY = m<xyz = XY = Z W 34 o 7 V Y X
M N Unit 2: Triangles and Quadrilaterals Lesson 2.7 Properties of Trapezoids and Kites Lesson 8.5 from textbook Objectives Use properties of trapezoids to find the measure of angles and sides of a trapezoid. Use the Midsegment Theorem for Trapezoids to find the measure of the parallel sides and midsegment of a trapezoid. Use properties of kites to find the measure of angles and sides of the kite. Trapezoid Isosceles Trapezoid Kite Isosceles Trapezoid Isosceles Trapezoid Isosceles Trapezoid Theorem #1 Theorem #2 Theorem #3 If a trapezoid is isosceles, then If a trapezoid has a pair of trapezoid is isosceles if each pair of base angles is congruent base angles, then it and only if its diagonals are congruent. is an isosceles trapezoid. congruent. Example 1 Find m<b, m<c, and m<d. 54 o > B m<b = m<d = m<c = D > C MIDSEGMENTS OF TRPEZOIDS The midsegment connects the midpoints of the of the trapezoid. CTIVITY: 1. Draw the midsegment of trapezoid MNOP and label it B. 2. Use your ruler and find PO, MN, and B. Write these measures on your diagram. P O
3. What is the connection between the midsegment B and the bases PO and MN? Midsegment Theorem for Trapezoids The midsegment of a trapezoid is parallel to each base and its length is. Example 2 B is the midsegment of the trapezoid. Find the value of x. 22 x = B x x - 2 43 B x = 30 50 Kite Theorem #1 Kite Theorem #1 If a quadrilateral is a kite, then its diagonals are perpendicular. If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. Example 3 B Find the lengths of the sides of the kite. B = BC = D = CD = 2 2 2 5 C D