Proving Theorems about Lines and Angles Lesson 6 Proving Theorems about Lines and Angles

Metalbro is a construction company involved with building a new skyscraper in Dubai. The diagram below is a rough sketch of a crane that Metalbro workers are using to build the skyscraper. The vertical line represents the support tower and the other line represents the boom. For safety reasons, the boom cannot be more than 15º beyond the horizontal in either direction. A horizontal line forms a 90º angle with the support tower. A straight line forms a 180º angle. Warm-Up

Essential Questions How do angle relationships work together in two pairs of intersecting, opposite rays? How do angle relationships work together in a set of parallel lines intersected by a transversal? How are angle relationships important in the real world? How do proofs apply to situations outside of mathematics?

Words To Know adjacent angles - angles that lie in the same plane and share a vertex and a common side. They have no common interior points. Alternate exterior angles - angles that are on opposite sides of the transversal and lie on the exterior of the two lines that the transversal intersects Alternate interior angles - angles that are on opposite sides of the transversal and lie within the interior of the two lines that the transversal intersects Complementary angles- two angles whose sum is 90º Corresponding angles - angles in the same relative position with respect to the transversal and the intersecting lines

Words To Know equidistant - the same distance from a reference point Exterior angles- angles that lie outside a pair of parallel lines Interior angles- angles that lie between a pair of parallel lines Linear pair - a pair of adjacent angles whose non-shared sides form a straight angle Non-adjacent angles - angles that have no common vertex or common side, or have shared interior points Perpendicular bisector - a line that intersects a segment at its midpoint at a right angle

Words To Know Perpendicular lines- two lines that intersect at a right angle (90˚), written as 𝐴𝐵 ⊥ 𝑃𝑄 . Line segments and rays can also be perpendicular. plane - a flat, two-dimensional figure without depth that is determined by three non-collinear points and extends infinitely in all directions Postulate – a true statement that does not require a proof Proof - if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent Right angle - an angle measuring 90˚

Words To Know Same-side exterior angles - angles that lie on the same side of the transversal and are outside the lines that the transversal intersects; sometimes called consecutive exterior angles Same-side interior angles - angles that lie on the same side of the transversal and are in between the lines that the transversal intersects; sometimes called consecutive interior angles Straight angle – an angle with rays in opposite directions; i.e., a straight line; a straight angle measures 180º. supplementary angles two angles whose sum is 180º Traversal - a line that intersects a system of two or more lines Vertical Angles - nonadjacent angles formed by two pairs of opposite rays

Key Concepts A straight line has a constant slope and parallel lines have the same slope. If a line intersects a set of parallel lines, then the angles in the same relative position have the same measures. Angles in the same relative position with respect to the transversal and the intersecting lines are corresponding angles. If the lines that the transversal intersects are parallel, then corresponding angles are congruent.

Key Concepts

Key Concepts Alternate interior angles are angles that are on opposite sides of the transversal and lie on the interior of the two lines that the transversal intersects. If the two lines that the transversal intersects are parallel, then alternate interior angles are congruent.

Key Concepts Same-side exterior angles are angles that lie on the same side of the transversal and are outside the lines that the transversal intersects. If the lines that the transversal intersects are parallel, then same-side exterior angles are supplementary. Same-side exterior angles are sometimes called consecutive exterior angles.

Key Concepts When the lines that the transversal intersects are parallel and perpendicular to the transversal, then all the interior and exterior angles are congruent right angles.

Example 1 Given 𝐴𝐵 ∥ 𝐷𝐸 , prove that ∆𝐴𝐵𝐶~∆𝐷𝐸𝐶.

Example 2 Given l∥𝑚, and line k is a traversal. Prove: ∠3≅∠6

Example 3 In the following diagram , 𝐴𝐵 ∥ 𝐶𝐷 and 𝐴𝐶 ∥ 𝐵𝐷 . If 𝑚∠1=3 𝑥+15 , 𝑚∠2=2𝑥+55, and 𝑚∠3=4𝑦+9, find the measures of the unknown angles and the values of x and y.

Example 4 In the following diagram , 𝐴𝐵 ∥ 𝐶𝐷 and 𝐴𝐶 ∥ 𝐵𝐷 . If 𝑚∠1=35°, 𝑚∠ 65°, find 𝑚∠𝐸𝑄𝐹

Example 5 When a person looks at an object, the light rays are refracted or distorted as they pass through the eye, and the image is transmitted upside down on the retina in the back of the eye. The object and its retinal image are similar. Prove that they are in proportion using similar triangles. A diagram is given below. Assume that both the person looking at the image and the image are vertical.