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Journal chapter 5

Journal chapter 5 . By: maria jose diaz-duran. Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and its converse. Perpendicular Bisector: is a line perpendicular to a segment at its midpoint.

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Journal chapter 5

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  1. Journal chapter 5 By: mariajosediaz-duran

  2. Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and its converse. • Perpendicular Bisector: is a line perpendicular to a segment at its midpoint. • Perpendicular Bisector Theorem: if a point is one the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. • Converse: if a point is equidistant fro the endpoints of a segment then it is on the perpendicular bisector of the segment.

  3. Examples: Proof:

  4. Describe what an angle bisector is. Explain the angle bisector theorem and its converse. • Angle Bisector: a ray that divides and angle into two congruent angles. • Angles Bisector Theorem: if a point is on the bisector of an angle then it is equidistant from the sides of the angle • Converse: if a point in the interior of an angle is equidistant from the sides of the angle then it is on the bisector of the angle.

  5. Examples: LM LM=JM LM= 12.8

  6. Describe what concurrent means. Explain the concurrency of Perpendicular bisectors of a triangle theorem. Explain what a circumcenter is. • Concurrent: when three or more lines intersect at one point. • Concurrency of Perpendicular Bisectors of a Triangle: is a point, a point where it intersects. • Circumcenter: is the point of concurrency.

  7. Examples:

  8. describe the concurrency of angle bisectors of a triangle theorem. Explain what an incenter is. • Concurrency of angle bisector: is the point were they intersect in an angle. • Incenter: is the center of the triangle inscribed circle, for example a triangle has three sides, so then it has three angle bisectors. So the angle bisectors of a triangle are also concurrent, that is called the incenter of the triangle.

  9. EXAMPLES:

  10. Describe what a median is. Explain what a centroid is. Explain the concurrency of medians of a triangle theorem. • Median: a segment whose endpoint are a vertex and the midpoint of the opposite side. • Concurrency of Medians of a Triangle Theorem: a segment whose endpoint are a vertex of the triangle and the midpoint of the opposite side.

  11. Examples:

  12. Describe what an altitude of a triangle is. Explain what an orthocenter is. Explain the concurrency of altitudes of a triangle theorem. • Altitude of a Triangle: a perpendicular segment from a vertex to the line containing the opposite side. • Concurrency of altitudes of a triangle theorem: if there is a perpendicular segment in a vertex then it contains a line in the opposite side.

  13. Examples:

  14. Describe what a midsegment is. Explain the midsegment theorem. • Midsegment: a segment that joins the midpoint of two side od the triangle. • Midsegment Theorem: a midsegment of a triangle is parallel to a side of the triangle and its length is half the length of that side.

  15. Examples:

  16. Describe the relationship between the longer and shorter sides of a triangle and their opposite angles • The relationship between the longer and shorter sides of a triangle are that if the legs have the same length or are equal and have the same measure then the angles opposite have also the same length. As we know the angles are complementary so then the length of the hypotenuse is the length of a leg twice. • Angle-Side Relationships: • If two sides of a triangles are not congruent, then the larger angle is opposite the longer side. • If two angles of a triangle are not congruent, then the longer side is opposite the larger angle.

  17. Examples:

  18. describe the exterior angle inequality • Exterior angle inequality: The measure of an exterior angle of a triangle is greater than the measure of the other interior angle.

  19. Examples:

  20. Describe the triangle inequality • Triangle Inequality: the sum of any two sides lengths of a triangle is greater that the third side length.

  21. Examples:

  22. describe how to write an indirect proof • Indirect Proof: you begin by assuming that the conclusion is false, then you show that this assumption leads to a contradiction, also called proof of contradiction.

  23. Examples: triangleLMNhas at most one right angle. 
Step 1: Assume triangleLMNhas more than one right angle. That is, assume that angle L and angle M are both right angles. 
Step 2: If M and N are both right angles, then m<L= m<M= 90  
Step 3: m<L+ m<M+ m<N= 180  The sum of the measures of the angles of a triangle is 180.
Step 4: Substitution gives 90 + 90 + m<N = 180.
Step 5: Solving gives m<N = 0.
Step 6: This means that there is no triangleLMN, which contradicts the given statement.
Step 7: So, the assumption that<L and <M are both right angles must be false.           
Step 8: Therefore, triangleLMNhas at most one right angle.

  24. real LIFE SITUATIONs: Sarah left her house at 9:00 AM and arrived at her aunt’s house 80 miles away at 10:00 AM. Use an indirect proof to show that Sarah exceeded the 55 mph speed limit. Indirect Proof: Sarah did NOT exceed the 55 mph speed limit. She drove 80 miles at 55 mph. At this speed, Sarah would need 80/55 (approximately) = 1 hour 45 minutes to reach her aunt’s place. But as per the problem she drove from 9:00 AM to 10:00 AM … exactly an hour. SO, she must have driven faster than 55 mph….a contradiction that Sarah did NOT exceed the speed limit. Therefore, Sarah exceeded the speed limit.

  25. Prove the following using an indirect proof. if 3n + 1 is even, then ‘n’ is odd. Indirect Proof: ‘n’ is NOT odd. ‘n’ is even. Then the given statement is: if 3n + 1 is even, then ‘n’ is EVEN” ‘n’ is even means ‘n’ is a multiple of 2. Then: 3n + 1 = 3(2) + 1 = 6 + 1 Well…6 is even. So, 6m + 1 is odd. Therefore, 3n + 1 is ODD…because 3n + 1 = 6 + 1 By assuming ‘n’ is even, we’ve shown that 3n + 1 is ODD which is a contradiction. Therefore: If ‘n’ is odd then 3n + 1 is even. This is the contrapositive of the statement to be proved. Since the contrapositive is true, it follows that the original statement "if 3n + 1 is even, then ‘n’ is odd" is true.

  26. Describe the hinge theorem and its converse • Hinge Theorem: if two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, that the longer third side is across from the larger included angle. • Converse: if two sides of one triangle are congruent to two sided of another triangle and the third sides are not congruent, then the larger included angle is across the longer third side.

  27. Examples:

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