Question

what are the values of x and y?
h
56°
e
x
g
100°
y
f
x =
°
y =
°

Explanation:

Step1: Use the property of congruent triangles

Since the two triangles are congruent (indicated by the equal - side markings), corresponding angles are equal. In a quadrilateral \(GHEF\) (composed of two congruent triangles), the sum of the interior angles of a quadrilateral is \((4 - 2)\times180^{\circ}=360^{\circ}\). Also, for congruent triangles, \(\angle G=\angle E + 2\times\angle\) (the non - equal angle in the congruent triangles situation). But more simply, for the congruent triangles: \(\angle E=\angle G\) is wrong. Wait, actually, for the two congruent triangles (by SSS - side - side - side congruence, as the side markings show three pairs of equal sides). \(\angle x=\angle G\) (corresponding angles of congruent triangles).

Step2: Calculate the value of \(y\)

We know that \(2x + 2y+56^{\circ}+ 100^{\circ}=360^{\circ}\). Since \(x = 100^{\circ}\) (from the congruent - triangle property, corresponding angles of congruent triangles are equal, \(\angle G\) and \(\angle E\) are corresponding angles of the two congruent triangles formed by the diagonal \(HF\)). Substitute \(x = 100^{\circ}\) into \(2x + 2y+56^{\circ}+ 100^{\circ}=360^{\circ}\).

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Another way: Since the two triangles are congruent (by SSS), \(\angle x=\angle G = 100^{\circ}\). And for the two congruent triangles, if we consider the sum of angles in one of the triangles. Let's assume the two congruent triangles \(\triangle GHF\) and \(\triangle EFH\). In \(\triangle GHF\), we know \(\angle G = 100^{\circ}\), and if we consider the other angles. But using the quadrilateral angle - sum formula \(360^{\circ}\). Let \(x\) and \(y\) be as defined. The two triangles are congruent, so \(x = 100^{\circ}\) (corresponding angles). Then \(2y+56^{\circ}=360^{\circ}-2\times100^{\circ}\)

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Answer:

\(x = 100\), \(y = 52\)