Warm Up Solve each proportion. 1. 3 = 15 3. x 20 2. x 35 3 4 24 4 y 7 1. 3 = 15 3. x 20 2. x 35 3 4 24 4 y 7 5

Dissections of Squares ` ACTIVITY 30min Materials Need glue & scissors Dissections of Squares Critical Thinking Cut out each figure (one at a time) and rearrange pieces to form a square. Glue pieces in the square provided.

Polygons and Similarities 6.1 and Chapter 7

6.1 Properties and Attributes of Polygons To identify and name polygons To find the sum of the measures of interior and exterior angles of convex polygons and measures of interior and exterior angles of regular polygons To solve problems involving angle measures of polygons

Define Polygons A closed figure Formed by segments

Polygons vs. Not Polygons

Regular Polygon Convex polygon where all sides and angles are congruent.

Types of Polygons A Convex polygon is a polygon such that no lines containing a side of the polygon contains a point in the interior of the polygon ~Angles face out A Concave polygon is a polygon for which there is a line containing a side of the polygon and a point in the interior of the polygon. ~ Angles cave in

Convex or Concave

Possible Diagonals/Triangles: Find the sum of degrees Determine how many triangles a shape have by drawing diagonals. Each triangle is 180 degrees.

Thm 6-1-1: Polygon Angle Sum Theorem Formula: S = 180 (n – 2) S= sum of the measures of interior angles N = number of sides Sum S = 180(n -2) S = 180(3-2) S = 180(1) S = 180 An interior angle I = S/n I = 180/3 I = 60

Thm6-1-2: Polygon Exterior Angle Sum Theorem The sum of the measures of the exterior angles is 360° To find one exterior angle use formula Formula: E = 360/ n E= measure of exterior angles N = number of sides E = 360/n E = 360/3 E = 120

Warm-Up: Polygons Number of sides: Name Polygon: Number of Triangles: Is polygon convex or concave. Find sum of the measures of interior angles:   Find the measure of an interior angle Find measure of an exterior angle: 6 Hexagon (n- 2) = 4 S = 180(n-2) 180(4)= 720 I = Sum n 720 = 120 6 360 = 60 6 E = 360 n

Twitter #polygonsproperties P. 397 Think and Discuss #1 Draw a concave pentagon and convex pentagon. Explain the difference between the two figures.

Measure of an Exterior Angle Textbook p. 395 Convex Polygons Number of Sides Number of triangles Sum of Angles Measures S = 180(n – 2) Measure of an Exterior Angle Ext = 360/n Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Hendecagon Dodecagon n - gon n (n – 2) S = 180( n – 2) Ext = 360 / n

Practice Geo: Textbook page 398 # 1- 15

Warm-Up: Polygons Number of sides: Name Polygon: Number of Triangles: Is polygon convex or concave. Find sum of the measures of interior angles:   Find the measure of an interior angle Find measure of an exterior angle: 7 Heptagon (n- 2) = 5 S = 180(n-2) 180(5)= 900 I = Sum n 900 = 128.57 7 360 = 51.43 7 E = 360 n

Warm Up: Part 2 Solve each proportion. 1. x = 11 2. 13 = 26 3. x-2 = 3 Cross Multiply - 35(x) = 11(5) - 35x = 55 35x = 55 35 35 - x = 1.57 Cross Multiply - 13(7x) = 49(26) - 91x = 1274 91x = 1274 91 91 - x = 14 Cross Multiply - 8(x-2) = 3(x) - 8x -16 = 3x 8x -8x -16 = 3x – 8x - -16 = -5x -16 = -5x -5 -5 - x = 3.2

Practice Geo: Textbook page 398 # 1- 15

Chapter 7: Connecting Proportion and Similarity Recognize and use ratios and proportions Identify similar figures and use the proportions of similar figures to solve problems Use proportional parts of triangles to solve problems

Exploring Similar Polygons Book Definition: Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. Symbol: In simpler terms: Two polygons with the same shape but are different sizes.

Are two congruent figures similar? Think about it…. Discuss… Yes, congruent figures have congruent angles and sides are proportional at 1:1 ratio

Scale Factor and Dilation Dilation is a transformation that reduces or enlarges figures On a camera: zoom-in/ zoom-out Scale Factor = ______ A 5 D 2.5 4 B 3 C E 10 H 5 dfgff 8 F 6 G 10 = 5 8 = 4 6 = 3 5 2.5

Scale Factor Scale Factor = _____ 4 8 3 12 24 9 Scale Factor = ______ A 6 D 2 5 B 4 C E 12 H 4 dfgff 8 F 7 G 12 = 4 9 = 3 24 8 12 = 6 8 = 5 7 = 4 4 2

Similar Triangles: Property 1 __________________________: if _____________________ of one triangle are ____________ to __________________________ of another triangle, then the triangles are ____________. Ex. Side-Side-Side Similarity the three sides proportional the three corresponding sides similar

Similar Triangles: Property 2 Angle - Angle Similarity _________________________: If the measures of ___________ of ___________ are __________________________________ then the triangles are __________. Ex. two angles one triangle congruent to 2 angles of the other similar

Similar Triangles: Property 3 Side-Angle-Side Similarity ________________________: if the measures of __________ of a triangle are ___________ to the measures of _________ of another triangle and the _________ angles are _________, then the triangles are ___________. Ex. two sides proportional two sides included congruent similar

Similarity Ratio:

Triangle Proportionality Theorem VW = VX WY XZ

Similarity Ratio: AE = AD = ED AC AB CB

Similarity Ratio: EC = ED = CD EA EB AB X

Solve for x and y. 49 x y+3 20 29 21

Warm- up: 1. Draw a convex heptagon. 2. Draw a concave nonagon 3. Sketch a regular hexagon. Sum of interior angles: An Interior angle: An Exterior angle: S = 180(6-2) = 720 I = 720/6 = 120 E = 360/6 = 60

Check Practice: Similar Figures Notes: Sides are proportional & angles are congruent. P. 520 # 5, 6, 12, 13, 16, 17, 19

Quiz Time 10 mins Absolutely NO TALKING during quiz. Take your time Good LUCK!

Practice Find a partner. Complete “How Can You Tell Which End of a Worm Is His Head?” (must complete in pairs). Each partner complete a side For each answer, write in letter of question. EACH SIDE IS DIFFERENT! Each partner must show work to receive FULL CREDIT. Use notebook paper

Chapter 7: Practice

Warm-Up 1. Use the Interior Angle Theorem to find measure of each angle. C B 3x-12 4x-14 A 7x +5 5x -15 D 5x E 2. Are two congruent figures similar? Explain. 3. Quadrilaterals ABCD and AEFG are similar. AE=22, AB=5x + 4, EF=8 and BC=2x-2. Solve for x. A E B G F D C

Warm -Up 16 22 11 m 3 8 4 4 x

Warm up 2. What is the difference between concave and convex polygons? 3. regular heptagon, find each. A. Sum of I = B. I= C. E= D. Sum of E= 1. Similar? Yes or No a. 4 6 1.3 1.8 1.4 7 b. 6 8 15 20 5 12.5

Constructions Materials: paper, ruler and protractor Construct each (use polygon chart) Regular Pentagon ~ sides 3 inches Regular Octagon ~ sides 2 inches Your Choice ~ Regular Polygon ~ make sides a reasonable measure To start… find the measure of one interior angle. Draw one measured side with ruler. Next use protractor to measure angle Then draw 2nd side Repeat steps.

Practice: Constructions Construct each (use polygon chart) Materials: paper, ruler and protractor Include the sum of interior angles and the measure of one interior angle Regular Pentagon ~ sides 3 inches Regular Octagon ~ sides 2 inches Your Choice ~ Regular Polygon (4+ sides) ~ make sides a reasonable measure Bonus: Your Choice ~ Regular Polygon (4+ sides) ~ make sides a reasonable measure To start… Draw one measured side with ruler. Next find the measure of one interior angle and use protractor to measure interior angle Then draw 2nd side Repeat steps.

Warm up 2. What is the difference between concave and convex polygons? 3. regular heptagon, find each. A. Sum of I = B. I= C. E= D. Sum of E= 1. Similar? Yes or No a. 4 6 1.3 1.8 1.4 7 b. 6 8 15 20 5 12.5

Bonus 1. Draw if needed. Joyce sighted to the top of a tree along a stake that she knew to be 3 feet high. If she is standing 2 feet from the stake and 18 feet from the tree, how high is the tree? 2. Find x. 2x+32 8x+5 10x+2 12x+3 4x+12 4x+2 3. The lengths of the sides of a triangle are 3, 4, and 6. If the length of the shortest side of a similar triangle is 5, find the lengths of its other two sides.