Math / Geometry

QuestionThe swan below is composed of several triangles. Use the given information and the figure to find each angle measure. Note: Figure not drawn to scale.
Given: $\triangle A B C$ is equilateral; $\overline{D E} \cong \overline{C E} \cong \overline{E F}$ loverline $\{\mathrm{DE}\}$ \{congloverlinc $\{\mathrm{CE}\}$ /congloverline $\{\mathrm{EF}\} ; \overline{K O} \cong \overline{F O}$ loverline $\{\mathrm{KO}\}$ \congloverline $\{\mathrm{FO}\} ; \overline{J N} \cong \overline{J O}$ loverline $\{\mathrm{JN}\}$ /congloverline $\{\mathrm{JO}\} ; \angle B C D \cong \angle B D C ; C G F \cong \angle C F G ; \angle H K N \cong$ $\angle H N K ; \triangle G C F \cong \triangle G K F \cong \triangle J H M ; \triangle K F H \cong \triangle K L H$
Triangular Swan Portfolio Print the document so that you can write on it and mark the pictureshowin congruent sides - (2 points)
1. $m \angle \mathrm{ABC}=$ 60
2. $m \angle B C A=$ 60 $\square$
3. $m \angle \mathrm{CAB}=$ 60 $\square$
4. $m \angle B C D=$ 70 $\square$
5. $m \angle \mathrm{BDC}=\mid 70$

Studdy Solution

STEP 1

1. $\triangle ABC$ is equilateral, meaning all its angles are $60^\circ$ .
2. $\overline{DE} \cong \overline{CE} \cong \overline{EF}$ , indicating $\triangle DCE$ and $\triangle ECF$ are isosceles.
3. $\overline{KO} \cong \overline{FO}$ , indicating $\triangle KOF$ is isosceles.
4. $\overline{JN} \cong \overline{JO}$ , indicating $\triangle JNO$ is isosceles.
5. $\angle BCD \cong \angle BDC$ , indicating $\triangle BCD$ is isosceles.
6. $\angle CGF \cong \angle CFG$ , indicating $\triangle GCF$ is isosceles.
7. $\angle HKN \cong \angle HNK$ , indicating $\triangle HKN$ is isosceles.
8. $\triangle GCF \cong \triangle GKF \cong \triangle JHM$ , indicating these triangles are congruent.
9. $\triangle KFH \cong \triangle KLH$ , indicating these triangles are congruent.

STEP 2

1. Determine angles in $\triangle ABC$ .
2. Determine angles in $\triangle BCD$ .
3. Use isosceles and congruent triangle properties to find other angles.

STEP 3

Determine angles in $\triangle ABC$ :
Since $\triangle ABC$ is equilateral, each angle measures $60^\circ$ .
$m\angle ABC = 60^\circ$ $m\angle BCA = 60^\circ$ $m\angle CAB = 60^\circ$

STEP 4

Determine angles in $\triangle BCD$ :
Given $\angle BCD \cong \angle BDC$ and $m\angle BCD = 70^\circ$ , it follows that:
$m\angle BDC = 70^\circ$
Since the sum of angles in a triangle is $180^\circ$ , we find $m\angle CBD$ :
$m\angle CBD = 180^\circ - m\angle BCD - m\angle BDC$ $= 180^\circ - 70^\circ - 70^\circ$ $= 40^\circ$

STEP 5

Use isosceles and congruent triangle properties to find other angles:
This step requires more information about the specific triangles and angles in the swan figure, which is not provided in the problem statement. Therefore, we cannot proceed further without additional details.
The angles determined so far are:
1. $m\angle ABC = 60^\circ$
2. $m\angle BCA = 60^\circ$
3. $m\angle CAB = 60^\circ$
4. $m\angle BCD = 70^\circ$
5. $m\angle BDC = 70^\circ$

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