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Transcript of Perpendicular bisector – is a line that goes through a segment cutting it into equal parts,...
![Page 1: Perpendicular bisector – is a line that goes through a segment cutting it into equal parts, creating 90°angles Perpendicular bisector theorem – if.](https://reader036.fdocuments.in/reader036/viewer/2022062321/56649dd05503460f94ac4bcc/html5/thumbnails/1.jpg)
Jose Pablo Reyes 10 – 5
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Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and its converse. Give 3 examples of each.
Perpendicular bisector – is a line that goes through a segment cutting it into equal parts, creating 90°angles
Perpendicular bisector theorem – if a segment is bisected by a perpendicular line then any point on the perpendicular bisector is equidistant to the end point of the segment
Converse – is a point is equidistant from the endpoint of a segment, then the point lies on the perpendicular bisector of the segment
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Perpendicular bisector – examples
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Describe what an angle bisector is. Explain the angle bisector theorem and its converse. Give at least 3 examples of each.
Angle bisector – line/ray that goes exactly through the half of an angle
Angle bisector theorem – if an angle is bisected by a line/ray then any point on that line is equidistant from both sides of the angle
Converse – if a given point in the interior of an angle is equidistant from the sides of the angle then it is on the bisector of the angle
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Angle bisector – examples
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Describe what concurrent means. Explain the concurrency of Perpendicular bisectors of a triangle theorem. Explain what a circumcenter is. Give at least 3 examples of each.
Concurrent – three or more lines that intersect at one point
Circumcenter – is the point of congruency in a triangle which is where the points intersect.
Concurrency of perpendicular bisectors of a triangle theorem – perpendicular bisectors of a triangle intersect in a point that is equidistant from the vertices
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Concurrent, circumcenter,– examples
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concurrency of perpendicular bisectors of a triangle theorem – examples
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Describe the concurrency of angle bisectors of a triangle theorem. Explain what an incenter is. Give at least 3 examples of each.
The concurrency of angle bisectors of a triangle says that the 3 bisectors are congruent
The incenter is the point of concurrency of the angle bisectors, always inside the triangle
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Concurrency of angle bisector of a triangle theorem and incenter
– examples
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Describe what a median is. Explain what a centroid is. Explain the concurrency of medians of a triangle theorem. Give at least 3 examples of each.
Median – is a segments in which the endpoints are a vertex of the triangle and the midpoint of the opposite side
Centroid – is the point of concurrency of the medians of a triangle.
The centroid of a triangle is 2/3 of the distance from each vertex to the midpoint of the opposite side
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Median, Centroid of a Triangle – Examples
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Describe what an altitude of a triangle is. Explain what an orthocenter is. Explain the concurrency of altitudes of a triangle theorem. Give at least 3 examples.
Altitude of a triangle – is a segment from one vertex to the opposite side so the segment is perpendicular to the side . There are 3 altitudes for any triangle, it can be inside, outside or in the triangle
Orthocenter – is the point where the three altitudes of a triangle intersect. In obtuse triangles the orthocenter is outside
Concurrency of altitudes of a triangle theorem – the three lines containing the altitudes of a triangle are congruent
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Altitude and orthocenter – examples
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Concurrency of altitudes of a triangle theorem – Examples
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Describe what a midsegment is. Explain the midsegment theorem. Give at least 3 examples.
Midsegment of a triangle – any segment that joins the midpoint of two sides of the triangle
Midsegment theorem – a midsegment of a triangle is parallel to the side of the triangle and the length is half the length of that side
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Midsegment theorem – examples
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Describe the relationship between the longer and shorter sides of a triangle and their opposite angles. Give at least 3 examples.
Angle side relation in a triangle, the side opposite to the biggest angle will always have the longest length, the side opposite to the smallest angle will have the smallest length
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Angle side relation – examples
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Describe the exterior angle inequality. Describe the triangle inequality. Give at least 3 examples.
Exterior angle inequality – the exterior angle is greater than the non-adjacent interior angles of the triangle
Triangle inequality – any two sides of a triangle must add up to more than the 3rd side
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Exterior angle inequality, triangle inequality – examples
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Describe how to write an indirect proof. Give at
least 3 examples.
Steps to write an indirect proof – 1. Assume that what you are trying to
prove is false2. Try to prove it using the same steps as
in normal proofs3. When you face a contradiction, you
have to prove the theory true
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Indirect proofs – examples
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Describe the hinge theorem and its converse. Give at least 3 examples.
Hinge theorem – if two sides of two different triangles are congruent and the angle between them is not congruent, then the triangle with the larger angle will have the longer 3rd side
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Hinge theorem – examples