GEOMETRYGEOMETRY Math 7 Unit 4

Standards Strand 4: Concept 1 PO 1. Draw a model that demonstrates basic geometric relationships such as parallelism, perpendicularity, similarity/proportionality, and congruence. PO 6. Identify the properties of angles created by a transversal intersecting two parallel lines PO 7. Recognize the relationship between inscribed angles and intercepted arcs. PO 8. Identify tangents and secants of a circle. PO 9. Determine whether three given lengths can form a triangle. PO 10. Identify corresponding angles of similar polygons as congruent and sides as proportional. Strand 4: Concept 4 PO 6. Solve problems using ratios and proportions, given the scale factor. PO 7. Calculate the length of a side given two similar triangles.

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The hardest part about Geometry

Point : a location in space : think about the tip of your pencil A ●A

Line all the points on a never-ending straight path that extends in all directions A B

Segment all the points on a straight path between 2 points, including those endpoints C D

Ray a part of a line that starts at a point (endpoint) and extends forever in one direction E F

Angle formed by 2 rays that share the same endpoint. The point is called the VERTEX and the rays are called the sides. Angles are measure in degrees. A C B 1 Vertex Side 70

Angle A C B 15°

Plane a flat surface without thickness extending in all directions Think: a wall, a floor, a sheet of paper A

Parallel Lines lines that never intersect (meet) and are the same distance apart A B C D ║

Perpendicular Lines lines that meet to form right angles BA C D

Intersecting Lines lines that meet at a point B A C D

Right Angle An angle that measures 90 degrees.

Straight Angle An angle that measures 180 degrees or 0. (straight line)

Acute Angle An angle that measures between 1 and 89 degrees

Obtuse Angle An angle that measures between 91 and 179 degrees

Complementary Angles Two or more angles whose measures total 90 degrees. 1 2

Supplementary Angles Two or more angles that add up to 180 degrees. 1 2

*****Reminders****** Supplementary Straight angle Complimentary Corner

Adjacent Angles Two angles who share a common side B A C D

Example 1 Estimate the measure of the angle, then use a protractor to find the measure of the angle.

Example 1 Angles 1 and 2 are complementary. If m 1 = 60 , find m  = 90  2 = 90  = 90  - 60  2 = 30 

Example 3 Angles 1 and 2 are supplementary. If m 1 is 114 , find m < 1 + < 2 = 180  < 2 = 180  - < 1 < 2 = 180   < 2 = 66  114 

7.2 Angle Relationships t

Vertical Angles Two angles that are opposite angles. Vertical angles are always congruent!  1   3  2   4

Vertical Angles Example 1: Find the measures of the missing angles 55  ? ? 125  t

PARALLEL LINES Def: line that do not intersect. Illustration: l m A B C D AB || CD l || m

Examples of Parallel Lines Hardwood Floor Opposite sides of windows, desks, etc. Parking slots in parking lot Parallel Parking Streets: Arizona Avenue and Alma School Rd.

Examples of Parallel Lines Streets: Belmont & School

Def: a line that intersects two lines at different points Illustration: Transversal t

Supplementary Angles/ Linear Pair Two angles that form a line (sum=180  ) t  5+  6=180  6+  8=180  8+  7=180  7+  5=180  1+  2=180  2+  4=180  4+  3=180  3+  1=180

Supplementary Angles/ Linear Pair Find the measures of the missing angles ? 72  ? t 108 

Alternate Exterior Angles Two angles that lie outside parallel lines on opposite sides of the transversal t 22   7 11  

Alternate Interior Angles Two angles that lie between parallel lines on opposite sides of the transversal t 33   6 44  

Corresponding Angles Two angles that occupy corresponding positions. Top Left t Top Right Bottom Right Bottom Left 11   5 22   6 33   7 44  

Same Side Interior Angles Two angles that lie between parallel lines on the same sides of the transversal t  3 +  5 = 180 44 +  6 =

List all pairs of angles that fit the description. a.Corresponding b.Alternate Interior c.Alternate Exterior d.Consecutive Interior t

Find all angle measures 1 67  3 t 113   113 

Example 5: find the m  1, if m  3 = 57  find m  4, if m  5 = 136  find the m  2, if m  7 = 84 

Algebraic Angles Name the angle relationship –Are they congruent, complementary or supplementary? –Complementary Find the value of x x + 36 = x = 54  = 90  x  36 

Example 2 Name the angle relationship –Vertical –Are they congruent, complementary or supplementary? Find the value of x 115  x  x = 115  

Example 3 Name the angle relationship –Alternate Exterior –Are they congruent, complementary or supplementary? Find the value of x 125  t 5x  5x = x = 25 

Example 4 Name the angle relationship –Corresponding –Are they congruent, complementary or supplementary? Find the value of x 2x + 1 t 151  2x = x = 75  2x + 1 =

Example 5 Name the angle relationship –Consecutive Interior Angles –Are they congruent, complementary or supplementary? Find the value of x 81  t 7x + 15 supp 7x = 84 7 x = 12 7x + 96 = x = 180

Example 6 Name the angle relationship –Alternate Interior Angles –Are they congruent, complementary or supplementary? Find the value of x 3x t 2x + 20  20 = x 2x + 20 = 3x - 2x

The World Of Triangles

Pick Up Sticks For each given set of rods, determine if the rods can be placed together to form a triangle. In order to count as a triangle, every rod must be touching corner to corner. See example below.

ColorsDoes it make a triangle? Y/N a. orange, blue dark green b. light green, yellow, dark green c. red, white, black d. yellow, brown, light green e. dark green, yellow, red f. purple, dark green, white g. orange, blue, white h. black, dark green, red Yes No Yes No Yes

Can you use two of the same color rods and make a triangle? Explain and give an example. Makes a triangleDoes not make a triangle  Now find five new sets of three rods that can form a triangle. Find five new sets of rods that will not make a triangle.

 Without actually putting them together, how can you tell whether or not three rods will form a triangle?

Triangles A triangle is a 3-sided polygon. Every triangle has three sides and three angles, which when added together equal 180°.

Triangle Inequality: In order for three sides to form a triangle, the sum of the two smaller sides must be greater than the largest.

Triangle Inequality: Examples: Can the following sides form a triangle? Why or Why not? A. 1,2,2B. 5,6,15 Given the lengths of two sides of a triangle, state the greatest whole-number measurement that is possible for the third. A.3,5B. 2,8

TRIANGLES Triangles can be classified according to the size of their angles.

Right Triangles A right triangle is triangle with an angle of 90°.

Obtuse Triangles An obtuse triangle is a triangle in which one of the angles is greater than 90°.

Acute Triangles A triangle in which all three angles are less than 90°.

Triangles Triangles can be classified according to the length of their sides.

Scalene Triangles A triangle with three unequal sides.

Isosceles Triangles An isosceles triangle is a triangle with two equal sides.

Equilateral Triangles An equilateral triangle is a triangle with all three sides of equal length. Equilateral triangles are also equilangular. (all angles the same)

The sum of the interior angles of a triangle is 180 degrees. Examples: Find the missing angle: 70  50  xx xx 42 

The sum of the interior angles of a quadrilateral is 360 degrees. Examples: Find the missing angle: xx 80  60  xx

7.5 NOTES Congruent and Similar Def’n - congruent – In geometry, figures are congruent when they are exactly the same size and shape. Congruent figures have corresponding sides and angles that are equal. Symbol:

EX. 1 A B C All corresponding parts are congruent so F D E

Similar Def’n – similar – Figures that have the same shape but differ in size are similar. Corresponding angles are equal. Symbol: ~

Example 2 AB C D EF ________________ ~ _________________

Example 3: Find the value of x in each pair of figures. Corresponding sides are equal so MIKE JOSH 16 ft B RO TS L 2x ft HS 62 in E I M K JO 3x + 32 in 2x = x = 8 ft x + 32 = 62 3x = x = 10 in

Example 4 Sketch both triangles and properly label each vertex. Then list the three pairs of sides and three pairs of angles that are congruent.

NOTES on Similar Figures/Indirect Measurement Recall that similar figures have corresponding angles that are CONGRUENT but their sides are PROPORTIONAL. Def’n – ratio of the corresponding side lengths of similar figures (a.k.a. SCALE FACTOR) – corresponding sides of congruent triangles are proportional. One side of the first triangle over the matching side on the second triangle.

EX. 1 The triangles below are similar. c) Find the measure of <VWU. S 6 in. TR 105  V 5 in. WU a) Find the ratio of the corresponding side lengths. b) Complete each statement. i.) ii.) iii.) 105 

EX. 2 Write a mathematical statement saying the figures are similar. Show which angles and sides correspond. B H A I J D K C ABCD~IHKJ

You can use similar triangles to find the measure of objects we can’t measure. Use a proportion to solve for x. Example If find the value of x. 30x = x = 8 ft

Example 2: 5x = 70 5 x = 14 mm

Example 3: A basketball pole is 10 feet high and casts a shadow of 12 feet. A girl standing nearby is 5 feet tall. How long is the shadow that she casts? 10x = x = 6 ft

Example 4: Use similar triangles to find the distance across the pond. 10x = x = 36 m

CIRCLES

Radius (or Radii for plural) The segment joining the center of a circle to a point on the circle. Example: OA

Chord A segment joining two points on a circle Example: AB

Diameter A chord that passes through the center of a circle. Example: AB

Secant A line that intersects the circle at exactly two points. Example: AB

Tangent A line that intersects a circle at exactly one point. Example: AB

Arc A figure consisting of two points on a circle and all the points on the circle needed to connect them by a single path. A B Example: AB (

Central Angle An angle whose vertex is at the center of a circle. G Q H Example: <GQH

Inscribed Angle An angle whose vertex is on a circle and whose sides are determined by two chords. M N T Example: <MTN

Intercepted Arc An arc that lies in the interior of an inscribed angle. M N T Example: MN (

Important Information An inscribed angle is equal in measure to half of the measure of its intercepted arc. M N T So the measure < MTN of is equal to ½ of the measure of MN (

EX. 1 Refer to the picture at the right. e) Give the measure of arc AB. a) Name a tangent: b) Name a secant: c) Name a chord: d) Name an inscribed angle: EF BD AD <ADB 54 B A C D E F 27 