Question

7-7. are the two triangles below congruent? make a sketch of both triangles, then determine all of the angle measures and all of the side lengths. if they are congruent, describe a sequence of rigid transformations that maps one onto the other. note: the triangles are not drawn to scale. homework help

Explanation:

Step1: Calculate the third - angle of the first triangle

The sum of angles in a triangle is \(180^{\circ}\). Let the first triangle have angles \(A = 90^{\circ}\), \(B = 44^{\circ}\). Then the third angle \(C=180-(90 + 44)=46^{\circ}\).

Step2: Calculate the third - angle of the second triangle

Let the second triangle have angles \(D = 90^{\circ}\), \(E = 44^{\circ}\). Then the third angle \(F=180-(90 + 44)=46^{\circ}\).

Step3: Check side - angle correspondence

For the first triangle, if we assume the sides: using trigonometric ratios (right - triangle trigonometry, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)).
In a right - triangle with one non - right angle \(\theta = 44^{\circ}\) and adjacent side \(a = 44\). Let the hypotenuse be \(h\). Then \(\cos44^{\circ}=\frac{44}{h}\), \(h=\frac{44}{\cos44^{\circ}}\approx60\). The opposite side \(o = h\sin44^{\circ}\approx41.6\).
In the second right - triangle with adjacent side to \(44^{\circ}\) angle \(a'= 125\). Let the hypotenuse be \(h'\). Then \(\cos44^{\circ}=\frac{125}{h'}\), \(h'=\frac{125}{\cos44^{\circ}}\approx170.5\). The opposite side \(o'=h'\sin44^{\circ}\approx118.2\).
Since the side lengths are not equal (e.g., if we assume the sides related to the \(44^{\circ}\) angle in the two right - triangles: if the triangles were congruent, the ratios of corresponding sides would be \(1\)).

Answer:

The two triangles are not congruent.