Question

1. which condition does not prove that two triangles are congruent? (1) sss = sss (2) ssa = ssa (3) sas = sas solve for x 2) 3) find m∠dtu. 4) find the distance between each pair of 5) (5, -5), (-5, 1)

Explanation:

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). For the triangle in problem 2, we have the equation \(20^{\circ}+20^{\circ}+(x + 150)^{\circ}=180^{\circ}\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(20+20+x + 150=180\), which simplifies to \(x+190 = 180\).

Step3: Solve for \(x\)

Subtract 190 from both sides of the equation: \(x=180 - 190=-10\).

Step4: For problem 3, use the exterior - angle theorem

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \(20x=-1 + 12x+34\).

Step5: Simplify the equation

Subtract \(12x\) from both sides: \(20x-12x=-1 + 34\), which gives \(8x=33\). Then \(x=\frac{33}{8}=4.125\).

Step6: Find \(m\angle DTU\)

\(m\angle DTU = 20x\), substituting \(x = 4.125\), we get \(m\angle DTU=20\times4.125 = 82.5^{\circ}\).

Step7: For problem 4, use the triangle angle - sum theorem

In the first triangle, the third angle is \(180-(41 + 49)=90^{\circ}\). Then, in the second triangle, using the angle - sum theorem \(180-(90 + 80)=10^{\circ}\).

Step8: For problem 5, use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

Here, \(x_1 = 5,y_1=-5,x_2=-5,y_2 = 1\). Then \(d=\sqrt{(-5 - 5)^2+(1+5)^2}=\sqrt{(-10)^2+6^2}=\sqrt{100 + 36}=\sqrt{136}=2\sqrt{34}\approx11.66\).

Answer:

1. \(x=-10\); 3) \(m\angle DTU = 82.5^{\circ}\); 4) The unknown angle is \(10^{\circ}\); 5) The distance \(d = 2\sqrt{34}\approx11.66\)