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Transcript
Unit 04 Triangles Vocabulary: triangle – a figure formed by three segments joining three non-collinear points Side classifications for triangles equilateral triangle – has three congruent sides isosceles triangle – has at least two congruent sides scalene – has no congruent sides angel classifications for triangles equiangular triangle – has three congruent angles acute triangle – has three acute angles right triangle – has one right angle obtuse triangle – has one obtuse angle interior angles – the three angles of a triangle exterior angles – when the sides of a triangle are extended, the angles that are adjacent to the interior angles Perpendicular bisector – segment, ray, line, or plane that is perpendicular to a segment at its midpoint. Equidistant from two point – if the point’s distance from each point is the same Distance from a point to a line – length of the perpendicular segment from the point to the line Equidistant from two lines – a point is the same distance from one line as it is from another Concurrent lines – when three or more lines intersect at the same point point of concurrency – the point of intersection of concurrent lines Medians – segment whose endpoints are a vertex and the midpoint of the opposite side Altitudes – perpendicular segment form the vertex to the opposite side of the triangle or the line that contains the opposite side Midsegment of a triangle – segment that connects the midpoints of two sides of a triangle Postulates, theorems, and corollaries: theorem 4.1 – Triangle sum theorem - The sum of the measures of the interior angles of a triangle is 180° Theorem 4.2 – Exterior angle theorem - the measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles Corollary of the triangle sum theorem - The acute angles of a right triangle are complementary Theorem 4.6 – Base angles theorem - if two sides of a triangle are congruent, then the angles opposite them are congruent Theorem 4.7 – converse of the base angles theorem - if two angles of a triangle are congruent, then the sides opposite them are congruent corollary to theorem 4.6 - if a triangle is equilateral, then it is equiangular corollary to theorem 4.7 - if a triangle is equiangular, then it is equilateral Theorem 5.1 – perpendicular bisector theorem - if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Theorem 5.2 – converse of the perpendicular bisector theorem - if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector Theorem 5.3 – angle bisector theorem - if a point is on the bisector of an angle, then it is equidistant from the sides of the angle Theorem 5.4 – converse of the angle bisector theorem - if a point is equidistant from the sides of an angle, then it is on the angle bisector theorem 5.5 - The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle Theorem 5.6 - The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle Theorem 5.7 - The medians of a triangle intersect at a point that is 2/3 of the distance form the vertex to the midpoint of the opposite side Theorem 5.8 - The lines containing the altitudes of a triangle are congruent Theorem 5.8 - The lines containing the altitudes of a triangle are congruent theorem 5.9 – Midsegment Theorem - the segment connecting the midpoints of two sides of a triangle (midsegment) is parallel to the third side and is half as long. 5.10 – if one side of a triangle is longer than another side then the angle opposite the longer side is larger than the angle opposite the shorter side 5.11 – if one angle of a triangle is larger than another, then the side opposite the larger angle is longer than the side opposite the smaller angle 5.12 – Exterior angle inequality – the measure of the exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles 5.13 – triangle inequality – the sum of the lengths of any two sides of a triangle is greater than the length of the third side. 5.14 – hinge theorem – if two sides of one triangle are congruent to two sides of another triangle, and the included angles of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second 5.15 – converse of the hinge theorem - if two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second then the included angles of the first is larger than the included angle of the second. Point of Concurrency Lines that are Concurrent perpendicular bisectors segment, ray, line, or plane that Circumcenter is perpendicular to a side at its (center of the midpoint circumscribed circle) Incenter (center of the inscribed circle) Centroid (center of balance) Orthocenter Finding Coordinates algebraically 1. 2. 3. 4. 5. 6. Find the midpoint of two side using the midpoint formula Find the slope of the same two sides using the slope formula for each side use its slope and midpoint to find the equation of the line perpendicular through the midpoint perpendicular slope (opposite reciprocal) y-intercept (b = y – mx) equation (y = mx + b) use substitution to solve for x. substitute x into either equation to find y (x , y) is the circumcenter Angle bisectors segment, ray, line, or plane that divides and angle into two congruent adjacent angles Medians segment whose endpoints are a vertex and the midpoint of the opposite side 1. 2. 3. 4. 5. 6. Altitudes perpendicular segment form the vertex to the opposite side of the triangle or the line that contains the opposite side Find the midpoint of two side using the midpoint formula Find the slopes using midpoint and opposite vertex use the slopes and midpoint to write the two equations slope from step 2 y-intercept (b = y – mx) equation (y = mx + b) use substitution to solve for x. substitute x into either equation to find y (x , y) is the centroid Theorems 5.5 The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle 5.6 The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle 5.7 The medians of a triangle intersect at a point that is 2/3 of the distance form the vertex to the midpoint of the opposite side 5.8 The lines containing the altitudes of a triangle are congruent