ABCD and CFGE are identical parallelograms. HCB and EDC are identical right-angled triangles. (a) Find ∠w. (b) Find ∠y. (c) Circle the words that describe BHF in the statement: BHF (is / is not) a straight line because the sum of angles BHC and CHF (is / is not) 180°.

Here's a step-by-step solution for the geometry problem: Part (a): Find w. Step 1: Identify properties of parallelogram ABCD. In parallelogram ABCD, opposite angles are equal, and consecutive angles sum to 180^. From the diagram, ADC = 80^. Therefore, BCD = 180^ - ADC = 180^ - 80^ = 100^. Step 2: Identify properties of parallelogram CFGE. The problem states that ABCD and CFGE are identical parallelograms. This means their corresponding angles are equal. Angle w is FGE. This angle corresponds to BCD in parallelogram ABCD. So, FGE = BCD = 100^. Alternatively, the diagram shows CGF = 80^. In parallelogram CFGE, FGE and CGF are consecutive angles, so they sum to 180^. w = FGE = 180^ - CGF = 180^ - 80^ = 100^ The value of w is 100^. Part (b): Find y. Step 1: Determine angles in parallelogram ABCD and CFGE. From Part (a), BCD = 100^. Since CFGE is identical to ABCD, its corresponding angles are equal. The angle FCE corresponds to DAB in parallelogram ABCD. In parallelogram ABCD, DAB = 180^ - ABC = 180^ - 80^ = 100^. So, FCE = 100^. Step 2: Determine angles in right-angled triangles HCB and EDC. The problem states that HCB and EDC are identical right-angled triangles. In HCB: CHB = 90^ (indicated by the square symbol) and HBC = 23^. The sum of angles in a triangle is 180^. HCB = 180^ - 90^ - 23^ = 67^ In EDC: CDE = 90^ (indicated by the square symbol) and DEC = 23^. DCE = 180^ - 90^ - 23^ = 67^ Step 3: Calculate y = HCF. From the diagram, the angles HCB, HCF, and FCE are arranged around point C. We can find HCD by subtracting HCB from BCD: HCD = BCD - HCB = 100^ - 67^ = 33^ Similarly, we can find DCF by subtracting DCE from FCE: DCF = FCE - DCE = 100^ - 67^ = 33^ Angle y is the sum of HCD and DCF: y = HCF = HCD + DCF = 33^ + 33^ = 66^ The value of y is 66^. Part (c): Circle the words that describe BHF in the statement: Step 1: Identify the known angles. For BHF to be a straight line, the sum of angles BHC and CHF must be 180^. From the diagram, BHC = 90^. Step 2: Calculate CHF. Consider HCF. We found HCF = y = 66^. From the diagram, HFC = 25^. The sum of angles in HCF is 180^. CHF = 180^ - HCF - HFC = 180^ - 66^ - 25^ = 180^ - 91^ = 89^ Step 3: Determine if BHF is a straight line. BHC + CHF = 90^ + 89^ = 179^ Since 179^ ≠ 180^, BHF is not a straight line. The completed statement is: BHF (is / is not) a straight line because the sum of angles BHC and CHF (is / is not) 180^. Got more? Send 'em!