Question

1. an art student is making a mosaic by gluing together triangular tiles with the dimensions shown. what is the value of x? 7. the structural frame for the roof of a garden shed has the dimensions shown. what is the value of x? 8. triangles abc and cde are shown. place an x in the table to show whether each congruence statement is true or false.

Explanation:

Problem 6

Step1: Identify congruent triangles

By SSS (Side - Side - Side) congruence criterion, the two triangles with sides \(2.4\) in, \(3\) in are congruent. So, the corresponding angles are equal.

Step2: Set up the equation

Since the angles are equal, \(6x - 8=46\)

Step3: Solve the equation

Add \(8\) to both sides: \(6x=46 + 8=54\)
Divide both sides by \(6\): \(x=\frac{54}{6}=9\)

Step1: Use the property of right - angled triangles in an isosceles triangle

The given triangle is isosceles (two sides \(6.5\) ft). The line perpendicular to the base bisects the vertex angle.

Step2: Use the angle sum property in a right - angled triangle

In a right - angled triangle, one angle is \(90^{\circ}\), another is \(33^{\circ}\). The third angle (half of \(x\)) is \(180-(90 + 33)=57^{\circ}\)

Step3: Find the value of \(x\)

Since \(x\) is the vertex angle of the isosceles triangle, \(x = 2\times57=114\)

For \(\overline{AB}\cong\overline{DE}\)

By Hypotenuse - Leg (HL) congruence (right - angled triangles with hypotenuse and one leg equal). In \(\triangle ABC\) and \(\triangle CDE\), if we assume \(AC = CE\) (marked as equal) and \(BC = CD\) (right - angled legs), then \(\overline{AB}\cong\overline{DE}\) (True)

For \(\triangle ABC\cong\triangle CDE\)

By HL congruence (right - angled triangles: \(\angle B=\angle D = 90^{\circ}\), \(AC = CE\), \(BC = CD\)). So \(\triangle ABC\cong\triangle CDE\) (True)

For \(\angle BAC\cong\angle DCE\)

Since \(\triangle ABC\cong\triangle CDE\), corresponding angles are equal. \(\angle BAC\) and \(\angle DCE\) are not corresponding angles. \(\angle BAC\cong\angle DEC\) (False)

For \(\angle ECD\cong\angle ACB\)

Vertical angles are equal. \(\angle ECD\) and \(\angle ACB\) are vertical angles (True)

Answer:

\(x = 9\)

Problem 7