Activity. Question. Materials. Explore. Think About It. Student Help. 1 On a piece of paper, draw a triangle and cut it out.
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1 ctivity 4.6 Intersecting edians Question What is the relationship between segments formed by the intersection of the medians of a triangle? aterials straightedge scissors ruler xplore 1 On a piece of paper, draw a triangle and cut it out. 2 Find the midpoint of each side by folding each side, vertex to vertex, and pinching the paper at the middle. raw a segment from each midpoint to the vertex of the opposite angle. hese segments are called medians. hink bout It 1. abel the triangle as shown above. opy and complete the table. tudent Help KI VIW For help with multiplying fractions, see p. 69. ength of median? F?? ength of segment from to vertex??? edian length 2 multiplied by 2? 2 F? 2? 2. Is there a relationship between the distance from to a vertex and the distance from that vertex to the midpoint of the opposite side?. xtension epeat teps 1 through with triangles of different shapes and sizes and complete a table like the one in xercise 1. o your results differ? xplain. 206 hapter 4 riangle elationships
2 4.6 edians of a riangle Goal Identify medians in triangles. Key Words median of a triangle centroid cardboard triangle will balance on the end of a pencil if the pencil is placed at a particular point on the triangle. Finding balancing points of objects is important in engineering, construction, and science. median of a triangle is a segment from a vertex to the midpoint of the opposite side. median median median X 1 raw a edian In, draw a median from to its opposite side. 4 6 olution he side opposite a is &*. Find the midpoint of &*, and label it. hen draw a segment from point to point. &* is a median of. 4 raw a edian opy the triangle and draw a median edians of a riangle 207
3 he following theorem tells you that the three medians of a triangle intersect at one point. his point is called the centroid of the triangle. HO 4.9 Intersection of edians of a riangle Words he medians of a triangle intersect at the centroid, a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. F ymbols If is the centroid of, then 2, 2 F, and 2. Itudent Help I ZO.O O X ore examples at classzone.com X 2 Use the entroid of a riangle is the centroid of and 27. Find and. olution Using heorem 4.9, you know that 2 2 (27) 18. ow use the egment ddition ostulate to find egment ddition ostulate ubstitute 27 for and 18 for ubtract 18 from each side. 9 implify. W &* has a length of 18 and &* has a length of 9. X Use the entroid of a riangle is the centroid of Q and 10. Find the length of &*. olution 2 Use heorem ubstitute 10 for. 2 (10) 2 2 ultiply each side by implify. W he median &* has a length of hapter 4 riangle elationships
4 Us th e troid f ria le he centroid of the triangle is shown. Find the lengths. 4. Find and,. Find JG and KG, 6. Find Q and, given 24. given JK 4. given Q 20. H J K F G 4.6 xercises Guided ractice Vocabulary heck In xercises 1 and 2, complete the statement. 1. he segment from a vertex of a triangle to the midpoint of the opposite side is a(n)?. 2. he? is the point where the three medians intersect.. opy, then draw the median from point to &* kill heck &* is a median of. Find the length of &* is the centroid of Q and 8. is the centroid of and. Find and. 12. Find and. 4.6 edians of a riangle 209
5 ractice and pplications xtra ractice ee p In xercises 9 and 10, copy the triangle and draw the three medians of the triangle. 9. F 10. K G J Using a entroid is the centroid of. Find and œ. 11. Q Q Q 0 Using a entroid is the centroid of. Find and rror nalysis is the centroid of. Your friend wants to find. he median 18. Find and correct the error. xplain your reasoning. 2 2 (18) 12 Homework H lp xample 1: xs. 9, 10, 18 xample 2: xs. 11 1, 17 xample : xs Finding a entroid raw a triangle and find the midpoint of each side. hen draw the three medians of your triangle. abel the centroid. 210 hapter 4 riangle elationships
6 Itudent Help I ZO.O HOWOK H xtra help with problem solving in xs is at classzone.com Using lgebra opy the graph shown. 19. Find the coordinates of Q,, and, the midpoints of the sides of the triangle. 20. Find the length of each median. 21. Find the coordinates of the centroid. abel this point as. 2 ( 1, 2) y (, 6) Π10 (11, 2) x tandardized est ractice 22. ultiple hoice In the figure shown, is the centroid of JK and K 6. What is the length of &&? K J 2. ultiple hoice In the figure shown, is the centroid of and 12. What is the length of V &*? F 8 G 18 H 24 J 6 V U W ixed eview Finding ngle easures in a riangle Find ma1. (esson 4.2) lgebra kills Writing quivalent Fractions Write two fractions equivalent to the given fraction. (kills eview, p. 66) edians of a riangle 211
7.4 Showing Triangles are
7. howing riangles are imilar: and oal how that two triangles are similar using the and imilarity heorems. ey ords similar polygons p. he triangles in the avajo rug look similar. o show that they are similar,
The hypotenuse and a leg of one triangle are congruent to the hypotenuse and a leg of the other triangle. THEOREM 5.2. right triangles, and
5.4 ypotenuse-eg ongruence heorem: oal se the ongruence heorem and summarize congruence postulates and theorems. ey Words hypotenuse p. 192 leg of a right triangle p. 192 he triangles that make up the
5.3 Proving Triangles are
0 1 1 10 5.3 roving riangles are ongruent: and Goal how triangles are congruent using and. ey Words vertical angles p. 75 alternate interior angles p. 121 Geo-ctivity 1 raw a segment 3 inches long. abel
Isosceles Triangles. leg. base
6 4 What ou ll Learn ou ll learn to identif and use properties of isosceles triangles. Isosceles riangles ecall from Lesson 5 that an isosceles triangle has at least two congruent sides. he congruent sides
1.3 Points, Lines, and Planes
1.3 oints, ines, and lanes oal Use postulates and undefined terms. ey Words undefined term point, line, plane postulate collinear, coplanar segment ray he legs of the tripod touch the table at three points.
Angle Bisectors of Triangles
6 What You ll Learn You ll learn to identify and use angle bisectors in triangles. ngle isectors of Triangles ecall that the bisector of an angle is a ray that separates the angle into two congruent angles.
11.5 Inscribed Angles and
age 1 of 6 11.5 Inscribed ngles and olygons Goal Use properties of inscribed angles. ey Words inscribed angle intercepted arc inscribed circumscribed n inscribed angle is an angle whose vertex is on a
7.5 Proportions and. Similar Triangles. Geo-Activity. Goal Use the Triangle Proportionality Theorem and its converse.
7. roportions and imilar riangles Goal Use the riangle roportionalit heorem and its converse. Ke Words midsegment of a triangle Geo-ctivit 1 raw a triangle. Label its vertices,, and. Make sure that each
Altitudes and Perpendicular Bisectors
6 2 hat ou ll Learn ou ll learn to identify and construct s and perpendicular bisectors in triangles. ltitudes and erpendicular isectors In geometry, an of a triangle is a perpendicular segment with one
6.3 Showing Quadrilaterals are
6.3 howing Quadrilaterals are arallelograms Goal how that a quadrilateral is a parallelogram. Key Words parallelogram p. 30 Geo-ctivity aking arallelograms ut two straws to form two congruent pairs. 2
Using Corresponding Parts of Congruent Triangles
4-4 Using orresponding arts of ongruent riangles ontent tandards G..5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. lso G..12 bjective o use triangle
Objectives To use relationships among sides and angles of parallelograms To use relationships among diagonals of parallelograms
6-2 roperties of arallelograms ontent tandards.o.11 rove theorems about parallelogram. s include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each
Proving Congruence ASA, AAS
roving ongruence, Vocabulary included side Use the ostulate to test for triangle congruence. Use the heorem to test for triangle congruence. are congruent triangles used in construction? he ank of hina
Bisectors in Triangles
5-3 isectors in riangles ontent tandard G..3 onstruct the inscribed and circumscribed circles of a triangle... Objective o identify properties of perpendicular bisectors and angle bisectors an you conjecture
EXERCISES Practice and Problem Solving
XI ractice and roblem olving or more practice, see xtra ractice. ractice by xample xample (page 224) In each diagram, the red and blue triangles are congruent. Identify their common side or angle.. K 2.
1.6 Angles and Their Measures
1.6 ngles and heir Measures oal Measure and classify angles. dd angle measures. ey Words angle sides and vertex of an angle measure of an angle degree congruent angles acute, right, obtuse, and straight
Bisectors, Medians, and Altitudes
isectors, Medians, and ltitudes Identify and use perpendicular bisectors and angle bisectors in triangles. Identify and use medians and altitudes in triangles. Vocabulary perpendicular bisector concurrent
Proof EXAMPLE EXAMPLE. Given:
4-7 hat ou ll earn o identify congruent overlapping triangles o prove two triangles congruent by first proving two other triangles congruent... nd hy o identify overlapping triangles in scaffolding, as
Objectives To use the AA Postulate and the SAS and SSS Theorems To use similarity to find indirect measurements
7-3 roving riangles imilar ontent tandards G..5 Use... similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.G.5 rove the slope criteria for parallel and
Problem 2. Got It? Proving Triangle Parts Congruent to Measure Distance. Proof
4-4 Using orresponding arts of ongruent riangles ommon ore tate tandards G-..5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. lso G-..12 1, 3 bjective
To prove two triangles congruent using the SSS and SAS Postulates. Are the triangles below congruent? How do you know? 6 B 4
4-2 riangle ongruence by SSS and SS ommon ore State Standards -SR..5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. P 1, P 3, P 4, P 7 Objective
Essential Question How can you measure and classify an angle?
0 1 1.5 easuring and onstructing ngles ssential Question ow can you measure and classify an angle? easuring and lassifying ngles Work with a partner. ind the degree measure of each of the following angles.
To recognize congruent figures and their corresponding parts
4-1 ongruent igures ontent Standard Prepares for G.SR.5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. Objective o recognize congruent figures
Work with a partner. Use dynamic geometry software. a. Construct ABC and DEF with the side lengths given in column 1 of the table below.
.3 roving riangle imilarity by and OMMO O Learning tandards HG-.. HG-..5 HG-G..5 HG-MG..1 OUIG VIL GUM o be proficient in math, you need to analyze situations by breaking them into cases and recognize
Work with a partner. Use dynamic geometry software.
10.4 Inscribed ngles and Polygons ssential uestion How are inscribed angles related to their intercepted arcs? How are the angles of an inscribed quadrilateral related to each other? n inscribed angle
4-3. Triangle Congruence by ASA and AAS. Content Standard. Essential Understanding You can prove that two triangles are congruent
4-3 riangle ongruence by and ontent tandard G..5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. bjective o prove two triangles congruent using
EXERCISES Practice and Problem Solving
XI ractice and roblem olving For more practice, see xtra ractice. ractice by xample lgebra Find the value of x in each parallelogram. xample (page 95. 5.. 0. 56 5. 80 6. 6 xample (page 95 lgebra Find the
Essential Question What are some properties of trapezoids and kites? Recall the types of quadrilaterals shown below.
7.5 Properties of Trapezoids and ites ssential Question What are some properties of trapezoids and kites? ecall the types of quadrilaterals shown below. Trapezoid Isosceles Trapezoid ite PV I OVI PO To
b. Move BC so that B is on the smaller circle and C is on the larger circle. Then draw ABC.
5.5 Proving Triangle ongruence by ssential uestion What can you conclude about two triangles when you know the corresponding sides are congruent? rawing Triangles Work with a partner. Use dynamic geometry
Essential Question What are the properties of parallelograms?
7. roperties of arallelograms ssential uestion What are the properties of parallelograms? iscovering roperties of arallelograms Work with a partner. Use dynamic geometry software. a. onstruct any parallelogram
5.4 Intersecting Medians
Investigating g Geometry TIVITY Use before Lesson 5.4 5.4 Intersecting Medians M T I LS cardboard straightedge scissors metric ruler Q U S T I O N What is the relationship between segments formed by the
6 segment from vertex A to BC. . Label the endpoint D. is an altitude of ABC. 4 b. Construct the altitudes to the other two sides of ABC.
6. Medians and ltitudes of Triangles ssential uestion What conjectures can you make about the medians and altitudes of a triangle? inding roperties of the Medians of a Triangle Work with a partner. Use
Naming Points, Lines, and Planes
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Key Concept Congruent Figures
4-1 ongruent igures ommon ore State Standards Prepares for G-SRT..5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. P 1, P 3, P 4, P 7 Objective
Essential Question How can you use congruent triangles to make an indirect measurement?
5.7 Using ongruent riangles ssential uestion How can you use congruent triangles to make an indirect measurement? easuring the Width of a iver IIUI H OI O OH o be proficient in math, you need to listen
Name Class Date. This shows that A corresponds to Q. Therefore, A Q. This shows that BC corresponds to RS. Therefore, BC RS.
Name lass ate eteaching ongruent igures iven, find corresponding parts using the names. rder matters. or example, or example, his shows that corresponds to. herefore,. his shows that corresponds to. herefore,.
To identify congruence transformations To prove triangle congruence using isometries
9-5 -0-1 ongruence ransformations ontent tandards G..7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent... lso G..6, G..8 bjective o identif congruence
Study Guide and Intervention
IO 1-1 tudy Guide and Intervention oints, Lines, and lanes ame oints, Lines, and lanes In geometry, a point is a location, a line contains points, and a plane is a flat surface that contains points and
2.3 Complementary and
2.3 omplementary and Supplementary ngles Goal Find measures of complementary and supplementary angles. ey Words complementary angles supplementary angles adjacent angles theorem Two angles are complementary
3. What is the simplified form of the Midpoint Formula if one of the endpoints of a segment is (0, 0) and the other is (x, y)?
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1-5. Skills Practice. Angle Relationships. Lesson 1-5. ALGEBRA For Exercises 9 10, use the figure at the right.
M IO kills ractice ngle elationships or xercises 6, use the figure at the right. ame an angle or angle pair that satisfies each condition.. ame two acute vertical angles. 2. ame two obtuse vertical angles.
To identify congruence transformations To prove triangle congruence using isometries
9-5 ongruence ransformations ommon ore tate tandards G-.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent... lso G-.B.6, G-.B.8 M 1, M 3, M bjective
To classify polygons in the coordinate plane
6-7 Polgons in the oordinate Plane ontent Standard G.GP.7 Use coordinates to compute perimeters of polgons... bjective o classif polgons in the coordinate plane ppl what ou learned - about classifing polgons.
Special Segments in a Circle
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5.4. Equilateral and Isosceles Triangles
OMMON OR Learning Standards HSG-O..10 HSG-O..13 HSG-MG..1.4 ONSRUING VIL RGUMNS o be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth
5 Congruent Triangles
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D AC BC AB BD m ACB m BCD. g. Look for a pattern of the measures in your table. Then write a conjecture that summarizes your observations.
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Triangle Relationships
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To use and apply properties of isosceles and equilateral triangles
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Chapter Quadrilaterals
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To draw and identify rotation images of figures
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7.3 Similar Right Triangles
Investigating g Geometr IVIY Use before Lesson 7. 7. imilar ight riangles MEIL rectangular piece of paper ruler scissors colored pencils Q U E I O N How are geometric means related to the altitude of a
Postulates and Diagrams
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Proving That a Quadrilateral Is a Parallelogram. To determine whether a quadrilateral is a parallelogram
- roving That a Quadrilateral Is a arallelogram ontent Standards G.O. rove theorems about parallelograms... the diagonals of a parallelogram bisect each other and its converse... lso G.ST. Objective To
Name Class Date. Given ABCD QRST, find corresponding parts using the names. Order matters.
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Quadrilaterals and Other Polygons
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Maintaining Mathematical Proficiency
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- DF is a perpendicular bisector of AB in ABC D
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2.7 Angles and Intersecting Lines
Investigating g Geometry TIVITY Use before Lesson.7.7 ngles and Intersecting Lines M T R I LS graphing calculator or computer Q U S T I O N What is the relationship between the measures of the angles formed
Solve each equation. EXAMPLE. Name &1 in two other ways. &AEC and &CEA are other names for &1. Quick Check
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MONITORING NG PROGRESS ANSWERS Chapter 5 Pacing Guide. 228 Chapter 5. Chapter Opener/ 0.5 Day Mathematical Practices. Chapter Review/ Chapter Tests
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Int. Geometry Unit 7 Test Review 1
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Properties of Rhombuses, Rectangles, and Squares
6- Properties of Rhombuses, Rectangles, and Squares ontent Standards G.O. Prove theorems about parallelograms... rectangles are parallelograms with congruent diagonals. lso G.SRT.5 Objectives To define
Essential Question How can you describe angle pair relationships and use these descriptions to find angle measures?
1.6 escribing Pairs of ngles OMMON OR Learning Standard HSG-O..1 ssential Question How can you describe angle pair relationships and use these descriptions to find angle measures? Finding ngle Measures
To name coordinates of special figures by using their properties
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Bisectors of Triangles
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