Question

14. In the diagram, BF I HD and GC 1 AE. Complete each statement. a) Ifm23 = 31°, then m25 =_? co C b) If m25 = 290, then M24 = 2. ? А 3 7. 4. c) If mZCAF = 122° then mZGAB =? 6 5 G E F d) Ifm27 35° then M23 =

Answer 1

a) Since the angles 3 and 4 sums 90, and the angles 4 and 5 sums 90 too, then we have

`\begin{gathered} m\angle3+m\angle4=31+m\angle4\text{ = 90,} \\ m\angle4=90-31 \\ m\angle4\text{ = 49.} \\ m\angle4+m\angle5=90 \\ m\angle5=90-m\angle4=90-49=31 \end{gathered}`

Then

`m\angle5=31\mathring{}`

b) Using the equations given above

`\begin{gathered} m\angle4+m\angle5=90 \\ m\angle4=90-m\angle5=90-29\text{ = 61} \\ m\angle4\text{ = 61 } \end{gathered}`

c) Note that the angle

`\begin{gathered} \angle CAF\text{ is composed by the angles }\angle3,\text{ }\angle4,\text{ and }\angle5, \\ \angle GAB\text{ is composed by }\angle2,\angle3,\angle4,\angle5,\angle6. \\ \text{Also, } \\ m\angle6=m\angle2\text{ (opposed angles by a vertex).} \end{gathered}`

So the angle

`\begin{gathered} m\angle2=180-122=58,\text{ } \\ \text{and then the angle } \\ m\angle GAB\text{ = 122+m}\angle2+m\angle6\text{ = 122+2(58) = }238\text{ } \\ m\angle GAB\text{ = 238 } \end{gathered}`

d)Since the angles 7 and 3 are opposed by a vertex, then their measurement is the same, so

`m\angle3=35\mathring{}`