Geometry Ch 1 Notes: Angles PDF

Geometry Ch 1 Notes: Angles DRHS For #1-4: Classify each angle as acute, obtuse, right, or straight. 1) 2) A) acute B) obtuse...

Geometry Ch 1 Notes: Angles DRHS For #1-4: Classify each angle as acute, obtuse, right, or straight. 1) 2) A) acute B) obtuse A) obtuse B) straight C) straight D) right C) right D) acute 3) 4) A) right B) obtuse A) acute B) straight C) straight D) acute C) obtuse D) right 5) 𝑚∠𝐴 = 3𝑥 − 24, and ∠𝐴 is a right angle. Find You try! 6) ∠𝐵 is a straight angle. Find x. the value of x. 9 Geometry Ch 1 Notes: Angles DRHS Angle Addition: For #7–8: Find the value of the missing angle for each diagram. 7) Find the value of b. 8) Find the 𝑚∠𝐴𝐷𝐵 if 𝑚∠𝐴𝐷𝐶 = 48. You Try! For #9–10: Find the value of the missing angle for each diagram. 9) Find the 𝑚∠𝑄𝑆𝑅. 10) Find y. 10 Geometry Ch 1 Notes: Angles DRHS 11) Find the measure of each angle. a. ∠𝐸𝐵𝐹 b. ∠𝐸𝐵𝐴 ° ° c. ∠𝐷𝐵𝐸 d. ∠𝐷𝐵𝐶 e. ∠𝐴𝐵𝐹 f. ∠𝐷𝐵𝐹 For #12-13: Find the value of the variable in each problem. 12) 𝑚∠𝑊𝑍𝑋 = 110° You try! 13) 11 Geometry Ch 1 Notes: Angles DRHS 1.2 Notes: Angle Pair Relationships Objectives: Students will identify angle pair relationships and use them to solve problems. ▪ Adjacent Angles ▪ Vertical Angles ▪ Linear Pairs Adjacent Angles Two angles are adjacent if they Definition of Adjacent share a common ray and Angles vertex. 1) For which diagrams below are angles 1 and 2 adjacent angles? a. b. c. Vertical Angles and Linear Pairs Two angles are vertical if they are Definition of non-adjacent angles formed by Vertical two intersecting lines. Angles Note: Their sides form opposite rays. Two adjacent angles form a Definition of linear pair if their non-common Linear Pair sides form a straight angle. 12 Geometry Ch 1 Notes: Angles DRHS 2) Determine if the following pair of angles are vertical angles, linear pairs, or neither. a. b. c. Exploration #1: Use this link to explore vertical angles’ relationship. Link: https://www.geogebra.org/m/SGhM48n5 Slide the rays into different positions and slide the shaded region into different positions. What do you think is true for any pair of vertical angles? (This is called a conjecture.) If two angles are congruent, then they have the __________ Congruent __________________. Angles Congruent symbol: If two angles form vertical Vertical Angle angles, then the two angles are Theorem _______________. 3) Find the variable for each diagram below. A) You Try! B) You Try! C) 13 Geometry Ch 1 Notes: Angles DRHS 4) Find the variable for each diagram below. A) B) You Try! C) You Try! D) Exploration #2: Use this link to explore the relationship with linear pairs. Link: https://www.geogebra.org/m/txA6R64k Slide the rays into different positions and note the measurements of the angles formed. What do you think is true for any linear pair? This is called a conjecture. If two angles form a linear pair, Linear Pair then the angles have a sum of Theorem ___________. 5) Find the variable for each diagram below. A) You Try! B) You Try! C) 14 Geometry Ch 1 Notes: Angles DRHS 6) Find the variable for each diagram below. A) You Try! B) C) You Try! D) 7) Find the measure of each variable in the diagrams below. A) B) C) Hint: Find x first! 15 Geometry Ch 1 Notes: Angles DRHS 1.3 Notes: More Angle Pair Relationships Objectives: Students will identify angle pair relationships and use them to solve problems. ▪ Complementary Angles ▪ Supplementary Angles ▪ Bisected Angles Exploration #1: Use the link below to explore complementary angles. Make sure you have chosen “complementary” on the drop-down menu. Link: https://www.visnos.com/demos/basic-angles Click on the rays and drag them to different positions. o Pay attention to the measures of the angles in the diagrams. Make a conjecture about complementary angles: Complementary Angles If two angles are complementary angles, then they have a sum of _________. Complementary Angles Note: Complementary angles do not have to be adjacent to each other. The complement of an angle is the degree The Complement of measure that adds up to _______ with the an Angle given angle measurement. 1) Which pairs of angles below are complementary angles? Select all that apply. A) 42° and 48° B) 20° and 160° C) 10° and 80 D) 90° and 90° E) 45° and 45° 2) Find the complement of each angle below, if possible. Remember, angle measures must be positive! A) 30° B) 71° C) 100° 16 Geometry Ch 1 Notes: Angles DRHS 3) ∠𝑨 is complementary to ∠𝑩. If 𝒎∠𝑨 = 𝟓𝟔°, then find 𝒎∠𝑩. You Try! 4) ∠𝑫 complementary to ∠𝑬. If 𝒎∠𝑫 = (𝟑𝒙 − 𝟐)°, and 𝒎∠𝑫 = (𝟑𝒙 − 𝟐)°, then find 𝒙. 5) Find x in each diagram below. A) You try! B) Exploration #2: Use the link below to explore supplementary angles. Make sure you have chosen “supplementary” on the drop-down menu. Link: https://www.visnos.com/demos/basic-angles Click on the rays and drag them to different positions. Pay attention to the measures of the angles in the diagrams. Make a conjecture about supplementary angles: Supplementary Angles If two angles are supplementary angles, then they have a sum of Definition of Supplementary ___________. Angles Note: Supplementary angles do not need to be adjacent to each other. The supplement of an angle is the Supplement of degree measure that adds up to an Angle _________ with the given angle measurement. 17 Geometry Ch 1 Notes: Angles DRHS 6) Which pairs of angles below are supplementary angles? Select all that apply. A) 42° and 48° B) 20° and 160° C) 10° and 80° D) 90° and 90° E) 45° and 45° 7) Find the supplement of each angle below, if possible. Remember, angle measurements must be positive values. A) 30° B) 71° C) 90° D) 132° E) 200° 8) Find the measure of ∠𝟑. 9) ∠1 and ∠2 are supplementary angles. ∠1 = (4𝑥 + 8)° and ∠2 = (𝑥 + 2)°. Find the value of x. Exploration #3: Click on the link below to explore an angle bisector. Link: https://www.geogebra.org/m/PrhX27f3 For this exploration, ⃡𝐷𝐵 bisects ∠ABC. Slide points 𝐴, 𝐵, 𝐶, and 𝐷 to different positions. What do you notice about the angles formed? Make a conjecture about what happens when an angle is bisected. 18 Geometry Ch 1 Notes: Angles DRHS Bisecting an Angle When an angle is bisected, Bisecting an two _________________ angles Angle are created. 10) In the diagram shown, ∠𝑃𝑄𝑅 is bisected by 𝑄𝑆. Find the value of x and the measure of ∠𝑃𝑄𝑅. 11) In the diagram shown, ∠𝑃𝑄𝑅 is bisected by 𝑄𝑆. The measure of ∠𝑃𝑄𝑅 is known to be 62°, Find the value of x and y. 12) Draw an angle that is bisected by a ray. Create measurements for all three angles in the diagram that verify that the angle is bisected. 19 Lines, line segments, and rays Lines, line segments, and rays Points, lines, line segments, and rays are the building blocks of geometry! What is a point? A point is an exact location or position. You can name a point using a letter. This point is named point A. A What is a line? A line is a collection of points in a straight path that goes on forever in both directions. You can name a line using two points on the line and a symbol with arrows pointing in both directions. This line is named AB. B A What is a line segment? A line segment is part of a line. It has two endpoints. You can name a line segment using its two endpoints and a symbol without arrows. This line segment is named AB. A B What is a ray? © 2024 IXL Learning. Visit IXL.com today! Page of - www.ixl.com Lines, line segments, and rays A ray is part of a line. It has one endpoint and continues forever in the other direction. You can name a ray using its endpoint, one other point on the ray, and a symbol with an arrow pointing in one direction. This ray is named AB. A B Go to IXL to try some practice problems! Points, lines, line segments, rays, and angles 9MK Visit IXL for more related skills and lessons! Skills Lessons Points, lines, line segments, rays, and angles 9MK Parallel and perpendicular lines Types of angles © 2024 IXL Learning. Visit IXL.com today! Page of - www.ixl.com Types of angles Types of angles What is an angle? An angle is formed by two rays that share a common endpoint, or vertex. The size of the angle depends on how widely or narrowly the two rays are spread apart. The wider open an angle is, the greater its measure. Angles are measured in degrees. Angles have special names based on their degree measures. Right angles A right angle measures exactly 90°. It is the same shape as the corner of a square. Acute angles An acute angle measures between 0° and 90°. It is narrower than a right angle. Obtuse angles An obtuse angle measures between 90° and 180°. It is wider than a right angle. © 2024 IXL Learning. Visit IXL.com today! Page of - www.ixl.com Types of angles Straight angles A straight angle measures exactly 180°. The rays form a line. Reflex angles A reflex angle measures between 180° and 360°. It is wider than a straight angle. Full angles A full angle measures exactly 360°. It is one complete rotation. © 2024 IXL Learning. Visit IXL.com today! Page of - www.ixl.com Types of angles Go to IXL to try some practice problems! Acute, right, obtuse, and straight angles R5K Visit IXL for more related skills and lessons! Skills Lessons Acute, right, obtuse, and straight angles R5K Lines, line segments, and rays Measuring angles with a protractor Types of triangles © 2024 IXL Learning. Visit IXL.com today! Page of - www.ixl.com Parallel and perpendicular lines Parallel and perpendicular lines What are parallel lines? Parallel lines are lines that always stay the same distance apart from each other. They will never meet. What are perpendicular lines? Perpendicular lines are lines that meet at right angles. What are intersecting lines? Intersecting lines are lines that meet or cross each other. They share a common point called the point of intersection. © 2024 IXL Learning. Visit IXL.com today! Page of - www.ixl.com Parallel and perpendicular lines Perpendicular lines intersect each other at 90° angles. Go to IXL to try some practice problems! Parallel, perpendicular, and intersecting lines 8VQ © 2024 IXL Learning. Visit IXL.com today! Page of - www.ixl.com Geometry Ch 1 Notes: Angles DRHS Topic 2 Notes: Simplifying Radicals, Naming Shapes, & Plotting Points Simplifying a radical: For #1–3: Simplify each radical expression (no decimal answers.) 1) √24 2) −3√50 3) √192 You try! For #4-6: Simplify each radical expression (no decimal answers). 4) 2√12 5) √54 6) 5√60 4 Geometry Ch 1 Homework DRHS Topic 2 Worksheet Name: ________________________ For #1-6: Simplify each rational expression (no decimal answers). Show your work. 2) √72 3) √63 1) √150 4) 2√8 5) √540 6) 5√24 For #7 – 15: name each shape in as many ways as possible. 7) the angle shown below 8) the segment shown below 9) the line shown below (4 ways) (2 ways) (3 ways) 10) the ray shown below 11) ∠1 as shown below 12) the ray shown below (2 ways) (2 ways) (2 ways) 13) the angle shown below 14) ∠4 as shown below 15) the line shown below (4 ways) (2 ways) (6 ways) Topic 2 Worksheet continued on next page… Geometry Ch 1 Homework DRHS Topic 2 Worksheet, continued. 16) Which options below are not valid ways to name ∠7? Select all that apply. A. ∠𝑀 B. ∠𝐿𝑀𝑃 C. ∠𝑁𝑀𝐿 D. ∠𝐿𝑃𝑀 E. ∠𝑃𝑀𝐿