Proving Theorems about Lines and Angles Angle Vocabulary Complementary- two angles whose sum is 90 degrees. Supplementary- two angles whose sum is 180 degrees. Congruent angles- two or more angles with the same measure. Angle bisector- a ray or a line segment that dividends an angle into two congruent angles. Vertical angles- are nonadjacent angles formed by two pairs of opposite rays. vertical angles are congruent. Linear pair- two adjacent angles whose non shared sides form a straight angle. Linear pairs are supplementary. Ex.1 angle one and two are complementary. Solve for x and the measure of both angles.
Ex.2 angle one and two are supplementary. Solve for x and the measure of both angles. Ex.3 Find angle three and four if...
Lines and Transversals Transversal- is a line that intersects a system of two or more lines. Two line are parallel if they do not intersect. Perpendicular lines are two lines that intersect at a right angle. Corresponding angles- angles with the same relative position with spect to the transversal and the intersecting lines. Corresponding angles are congruent Alternate interior angles- are on opposite sides of the transversal and lie on the interior of the two lines that the transversal intersect. Alternate interior angles are congruent. Same side interior angles- are angles that lie on the same side of the transversal and are in between the lines that the transversal intersects.
Same side interior angles are supplementary. Alternate exterior angles- are angles that are on the opposite sides of the transversal and lie on the exterior of the two lines that the transversal intersects. Alternate exterior angles are congruent. Perpendicular Transversal Theorem- if a transversal is perpendicular to one of the two parallel lines, then it is perpendicular to the other.
Ex.1 Ex.2 Ex.3
Ex. 4
Proving Theorems about Triangles Triangle Sum Theorem- the sum of the angle measures of a triangle is 180 degrees. Scalene triangle- no congruent sides. Isosceles triangles- two congruent sides. Equilateral triangles- three congruent sides. Exterior Angle Theorem- the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.
Ex.1 Find angle C. Ex.2 Find the missing angles.
Equiangular- all angles are congruent. If a triangle is equilateral then it's equiangular. If a triangle is equiangular, then it is equilateral. Ex.1 Find the measure of each angle.
Ex.2 Find the values for x and y.
Midsegment- is a line segment that joins the midpoints of two sides of a triangle. Triangle Midsegment Theorem- A Midsegment of a triangle is parallel to the third side and is half as long. Ex.1 Find BC and YZ and angle AXZ.
Properties of parallelograms Quadrilateral- is a polygon with four sides. The sum of all the angles is 360 degrees. Parallelogram- is a special type of quadrilateral with two pairs of opposite sides that are parallel. Opposite angles and sides are congruent. Ex.1 Find the missing angles of parallelogram ABCD if angle A is 65 degrees.
Ex.2 Find angle G
Trapezoids Trapezoid has exactly one pair of opposite parallel sides. Isosceles trapezoid- has one pair of opposite parallel sides and congruent legs. The median is one half the sum of the bases Consecutive angles are supplementary Ex.1 Find EF and angle A.
Ex.2 Find BD, angle B, and angle D. Polygons A polygon is a closed figure with three or more sides. A regular polygon is a polygon with all equal sides.
Convex polygon- is a polygon with no interior angles greater than 180 degrees and where all diagonals lie inside the polygon. Concave polygon- is a polygon with at least one interior angle greater than 180 and are at least one diagonal that does not lie entirely inside the polygon. The sum interior angles of a regular polygon can be found by multiplying the number of triangles by 180. Exterior angle can be found by extending only one of its sides. Ex.1 Find the sum of the interior angles and one interior angle.
Ex.2 Find one exterior angle of the polygon.
Angle Angle (AA) Similarity Is one statement that allows us to prove triangles are similar. Ex.1 Ex.2 Find DF
Ex.3 Solve for x.
Ex.4 Find x
Side Angle Side (SAS) If the measures of two sides of a triangle are proportional to the measures of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar. Side Side Side (SSS) If the measures of the corresponding sides of two triangles are proportional, then the triangles are similar. Ex.1 Prove the triangles are similar
Ex.2 Determine whether the triangles are similar.
Ex.3 Determine whether the triangles are similar. Ex.4 AB and DC are corresponding sides, and AE and DF are corresponding sides. Find x.
Proving Similarity Triangle Proportionality Theorem- If a line parallel to one side of the triangle intersects the other two sides of the triangle, then the parallel line divides these two sides proportionally. Triangle Bisector Theorem- if one angle of a triangle is bisected, or cut in half, then the angle bisector of the triangle divides the opposite side of the triangle into two segments that are proportional to the other two sides of the triangle.
Ex.1Find BE.
Ex.2 Find CA using two methods.
Ex.3 Prove DE // AC. Ex.4 Is DE//AC?
Ex.5 Find BD and DC.
Congruent Triangles If two or more triangles are proven congruent, then all of their corresponding parts are congruent. Criteria for Congruence side-side-side (SSS) Side-angle-side (SAS)
Angle-side-angle (ASA) Angle-angle-side (AAS) Hypotenuse-Leg (HL)
Ex.1 Determine which congruence statement can be used for the triangles.
Two Column Proofs Reflexive property: any quantity is equal to itself, example a=a. Ex.1 Given:
Ex.2 Given:
Ex.3 Given: