Question

1. an obtuse triangle is one that contains an angle larger than 90°. the triangle shown below is obtuse because m∠g = 110°. construct triangle mnp such that △mnp≅△fgh. use your protractor to verify that the measure of ∠n is 110°. leave all construction marks.
2. an equilateral triangle is a triangle whose three sides have the same length. use $overline{ab}$ as one side of the equilateral triangle. use a compass and straightedge to locate a point c so that △abc is equilateral.
3. it is known that △abc≅△def. if ab = 5, bc = 10, ac = 8, and df = 2x - 6, then solve for the value of x. it would be helpful to draw two congruent triangles and carefully label the three vertices on each.

reasoning

1. the two triangles shown below have the same angles, and hence the same “shape.” verify this fact by using tracing paper to experiment with laying the angles on top of each other. even though these two triangles have the same shape, they are not congruent. why?

Explanation:

Step1: Recall congruent - triangle property

Since \(\triangle ABC\cong\triangle DEF\), corresponding sides are equal. Side \(AC\) corresponds to side \(DF\).

Step2: Set up the equation

We know that \(AC = DF\). Given \(AC = 8\) and \(DF=2x - 6\), we set up the equation \(2x-6 = 8\).

Step3: Solve the equation for \(x\)

Add 6 to both sides of the equation: \(2x-6 + 6=8 + 6\), which simplifies to \(2x=14\). Then divide both sides by 2: \(\frac{2x}{2}=\frac{14}{2}\), so \(x = 7\).

Answer:

\(x = 7\)