Question

read questions 13-14 below carefully. a quality proof consisting of all annotations and appropriate statements and reasons must be created in order to receive full credit. #13.) (5 points) triangles cat and dog are congruent. if the m∠c = (x² − 6x)°, m∠t = (4x + 20)°, m∠o = (40°), determine the m∠hdg. #14.) (6 points) construct a two - column proof table for the information below. given: fg || kl, and fl bisects gk. prove: lk ≅ fg

Explanation:

Step1: Identify Congruent Triangles

Since \(\triangle CAT \cong \triangle DOG\), corresponding angles are equal. So \(\angle A = \angle O = 40^\circ\), and the sum of angles in a triangle is \(180^\circ\). Thus, \(\angle C + \angle T + \angle A = 180^\circ\). Substituting values: \((x^2 - 6x) + (4x + 20) + 40 = 180\).

Step2: Solve Quadratic Equation

Simplify the equation: \(x^2 - 6x + 4x + 20 + 40 = 180\) → \(x^2 - 2x + 60 = 180\) → \(x^2 - 2x - 120 = 0\). Factor: \((x - 12)(x + 10) = 0\). Solutions: \(x = 12\) or \(x = -10\). Since angle measure can't be negative, \(x = 12\).

Step3: Find \(\angle C\) and \(\angle T\)

For \(x = 12\), \(\angle C = 12^2 - 6(12) = 144 - 72 = 72^\circ\), \(\angle T = 4(12) + 20 = 48 + 20 = 68^\circ\). Check: \(72 + 68 + 40 = 180\), correct.

Step4: Determine \(\angle HDG\)

Since \(\triangle CAT \cong \triangle DOG\), \(\angle D = \angle A = 40^\circ\)? Wait, no—wait, \(\angle HDG\) is supplementary? Wait, no, \(\angle HDG\) corresponds to \(\angle C\) or \(\angle T\)? Wait, re - check: \(\triangle DOG\) has \(\angle O = 40^\circ\), \(\angle G=\angle T\), \(\angle D=\angle C\). Wait, \(\angle HDG\) is equal to \(\angle C\) (since \(HD \parallel\) maybe? Wait, the diagram: \(H - D - O\) is a line? Wait, no, \(\angle HDG\) is an angle at \(D\), and since \(\triangle DOG\) has \(\angle D=\angle C = 72^\circ\)? Wait, no—wait, when \(x = 12\), \(\angle C = 72^\circ\), so \(\angle D=\angle C = 72^\circ\)? Wait, no, \(\triangle CAT\): angles are \(\angle C = 72^\circ\), \(\angle T = 68^\circ\), \(\angle A = 40^\circ\). So \(\triangle DOG\): \(\angle O = 40^\circ\), \(\angle G = 68^\circ\), \(\angle D = 72^\circ\). Then \(\angle HDG\) is equal to \(\angle D\)? Wait, maybe \(\angle HDG\) is equal to \(\angle C\) or \(\angle T\)? Wait, no—wait, the problem says "determine \(m\angle HDG\)". Since \(\triangle DOG\) and \(\triangle CAT\) are congruent, \(\angle D=\angle C = 72^\circ\)? Wait, no, \(\angle D\) in \(\triangle DOG\) is equal to \(\angle C\) in \(\triangle CAT\) (corresponding angles). So \(m\angle HDG=\angle D = 72^\circ\)? Wait, no, wait: sum of angles in \(\triangle DOG\): \(\angle D+\angle O+\angle G = 180\). \(\angle O = 40\), \(\angle G=\angle T = 68\), so \(\angle D = 180 - 40 - 68 = 72\). So \(m\angle HDG = 72^\circ\).

Answer:

\(72^\circ\)