Geometry Chapter 5 Relationships Within Triangles

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1 Objectives: Section 5.1 Section 5.2 Section 5.3 Section 5.4 Section 5.5 To use properties of midsegments to solve problems. To use properties of perpendicular bisectors and angle bisectors. To identify properties of perpendicular bisectors. To identify properties of medians and altitudes of a triangle. To write the negation of a statement and the inverse and contrapositive of a conditional statement. To use indirect reasoning. To use inequalities involving angles of triangles. To use inequalities involving sides of triangles. Theorems and Postulates: Theorem 5-1: If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side, and is half its length. Theorem 5-2: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Theorem 5-3: If a point is equidistant from the endpoint of a segment, then it is on the perpendicular bisector of the segment. Theorem 5-4: If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. Theorem 5-5: If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector. Theorem 5-6: The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. Theorem 5-7: The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. Theorem 5-8: The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. Theorem 5-9: The lines that contain the altitudes of a triangle are concurrent. Corollary to the Triangle Exterior Angle Theorem: The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles. m1 m2 and m1 m 3 Theorem 5-10: If two sides of a triangle are not congruent, then the larger angle lies opposite the longer sides. If XZ XY, then my m Z

2 Theorem 5-11: If two angles of a triangle are not congruent then the longer side lies opposite the larger side. If ma mb, then BC AC. Theorem 5-12: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. XY YZ XZ YZ ZX YX ZX XY ZY Vocabulary: Midsegment Coordinate Proof Circumcenter of a triangle Circumscribed Incenter of a triangle Median of a triangle Centroid Orthocenter of a triangle Altitude of a triangle Inverse Contrapositive Equivalent Statements Indirect Reasoning Indirect Proof Daily Work 5 pts. each Homework 5.1 p #1-19odd, 22-25, 26, 29, p #1-4, 6, 12-15, 19-25, p #7, 8-11, 17, 29, 42, p #1-2, 3-15odd, 19-22all, 27,29, 31 Checkpoint Quiz p. 263 #1-10 (Check answers in the back of your book.) p #1-15odd, 17-19, 23, 29, 31-35odd 5.5 p #1-27odd Triangle Inequality Worksheet Review 1 p. 694 #1-30 Review 2 p. 281 #1-10, 11-41odd Points Earned

3 Journal (5 points per Question) Vocabulary (15 points) Chapter Project (30 points) Answer each of the following using complete sentences. 1. Draw a right triangle and its midsegments. Compare the four triangles created by the midsegments with the original triangle. Make a conjecture about your observations. 2. Explain and give an example of the difference between a median and a bisector of a triangle. 3. Give an example of a time where the median and the bisector are the same. 4. Use a conditional statement to write the inverse and the contrapositive of the statement. 1. Make a PowerPoint of 5 vocabulary terms, include definitions and pictures. 2. Make a poster of 5 vocabulary terms, include definitions and pictures. 3. Make vocabulary note cards for 10 of the vocabulary words with the word on one side and a picture and definition on the reverse. Attached is the information for the Chapter 5 Project. We will work on it for the first few days of Chapter 5. Points Earned

4

5 Investigating Special Segments of a Triangle Name Hour Date Terms you will need: Perpendicular bisector of a triangle Angle bisector of a triangle Median Altitude Your group will need to work together to do the following: Cut four large acute scalene triangles out of paper (use a ruler and protractor). Label the vertices of each triangle A, B, and C and number the triangles 1-4. For triangle #1 fold the triangle to produce the perpendicular bisectors of all three sides. For triangle #2 fold the triangle to produce the angle bisectors of all three angles. For triangle #3 fold the triangle to produce all three medians. For triangle #4 fold the triangle to produce all three altitudes. You may discuss the following questions with your group, but everyone needs to write the answers individually. Please use complete sentences. You will be investigating the results of the four folded triangles. Use a ruler and protractor to measure things and feel free to draw on your triangle with a compass. When you are measuring or drawing on your triangle you may find it handy to tape or glue it to another sheet of paper. Look at triangle #1. What do you observe to be true about the perpendicular bisectors of the sides of your triangle? Look for three things.

6 Look at triangle #2. What do you observe to be true about the angle bisectors of your triangle? Look for three things. Look at triangle #3. What do you observe to be true about the medians of your triangle? Please list all the things you can find. Look at triangle #4. What do you observe to be true about the altitudes of your triangle? Please list all the things you can find.

7 Build a Mobile Goal: Work with a partner to create a mobile with triangles that demonstrates how an object can be balanced at its centroid (point where the three medians of the triangle intersect). Your mobile should be designed to meet the following requirements: Consist of at least three levels. Contain at least two pairs of congruent triangles (make it clear which are congruent). Include a right triangle, equilateral triangle, isosceles triangle, and a scalene triangle. Balanced so it moves in a gentle breeze or when you touch it. Neat, well made with attention to details. It should be interesting to look at. Materials you will have available to you: String Sticks of some kind Thin cardboard Protractors Rulers Compasses How you will be graded: Levels within your mobile Congruent triangles Right, equilateral, isosceles, and scalene triangles Balance Craftsmanship Total Possible 3 points 2 points 1 point 0 points All 3 levels present 2 levels 1 level Nothing functional Both pairs of congruent triangles present and congruent. All for present and accurately made. All three levels balance and move when lightly tapped. Neat, well made and interesting to look at. Use of pattern and/or color. 15 points Either one pair of triangles perfectly congruent or 2 pairs with minor errors. Three of the four present and accurately made or all four made with a few minor errors. Two levels balance and move when lightly tapped. Two out of neat, well made, or interesting. Attempted to make one pair of congruent triangles, but they contain errors. At least two attempted with only minor errors or 3-4 with more errors. One level balances and moves when lightly tapped. One out of neat, well made, or interesting. Not attempted. Not attempted. Not Balanced. A mess.

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