Math  /  Geometry

QuestionTriangular Swan Portfolio The 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: ABC\triangle A B C is equilateral; KOFO,JNJODECEEFBCDBDCK O \cong F O, J N \cong J O D E \cong C E \cong E F \quad \angle B C D \cong \angle B D C; CGFCFG;HKNHNK;GCFGKFJHM;KFHKLH\angle 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
1. mABCm \angle A B C
9. mCEDm \angle C E D
17. mKFGm \angle \mathrm{KFG}
25. mHIMm \angle \mathrm{HIM}
33. mOKNm \angle O K N
18. mCKFm \angle C K F
26. mHMIm \angle \mathrm{HMI}
34. mLINO

Studdy Solution

STEP 1

1. ABC\triangle ABC is equilateral, meaning all its angles are 6060^\circ.
2. Congruent segments and angles are given, which implies equal measures.
3. Congruent triangles have corresponding angles and sides equal.
4. The sum of angles in a triangle is 180180^\circ.

STEP 2

1. Determine the measure of ABC\angle ABC.
2. Determine the measure of CED\angle CED.
3. Determine the measure of KFG\angle KFG.
4. Determine the measure of HIM\angle HIM.
5. Determine the measure of OKN\angle OKN.
6. Determine the measure of CKF\angle CKF.
7. Determine the measure of HMI\angle HMI.
8. Determine the measure of LINO\angle LINO.

STEP 3

Since ABC\triangle ABC is equilateral, each angle is:
mABC=60 m \angle ABC = 60^\circ

STEP 4

Given BCDBDC\angle BCD \cong \angle BDC and BCD\triangle BCD is isosceles, let each base angle be xx. Then:
2x+CBD=180 2x + \angle CBD = 180^\circ
Since ABC=60\angle ABC = 60^\circ, CBD=60\angle CBD = 60^\circ. Thus:
2x+60=180 2x + 60^\circ = 180^\circ 2x=120 2x = 120^\circ x=60 x = 60^\circ
Therefore, mCED=60 m \angle CED = 60^\circ because CED\angle CED is part of CDE\triangle CDE which is congruent to BCD\triangle BCD.

STEP 5

Given GCFGKF\triangle GCF \cong \triangle GKF, and CGFCFG\angle CGF \cong \angle CFG, each angle in GCF\triangle GCF is:
mKFG=60 m \angle KFG = 60^\circ

STEP 6

Since JHM\triangle JHM is congruent to GCF\triangle GCF, each angle in JHM\triangle JHM is:
mHIM=60 m \angle HIM = 60^\circ

STEP 7

Given HKNHNK\angle HKN \cong \angle HNK, and HKN\triangle HKN is isosceles, each angle is:
mOKN=60 m \angle OKN = 60^\circ

STEP 8

Since KFHKLH\triangle KFH \cong \triangle KLH, each angle in KFH\triangle KFH is:
mCKF=60 m \angle CKF = 60^\circ

STEP 9

Since HMI\triangle HMI is congruent to JHM\triangle JHM, each angle in HMI\triangle HMI is:
mHMI=60 m \angle HMI = 60^\circ

STEP 10

Assuming LINO\angle LINO is part of a congruent triangle similar to those previously discussed, it is:
mLINO=60 m \angle LINO = 60^\circ
The measures of the angles are:
1. mABC=60 m \angle ABC = 60^\circ
9. mCED=60 m \angle CED = 60^\circ
17. mKFG=60 m \angle KFG = 60^\circ
25. mHIM=60 m \angle HIM = 60^\circ
33. mOKN=60 m \angle OKN = 60^\circ
18. mCKF=60 m \angle CKF = 60^\circ
26. mHMI=60 m \angle HMI = 60^\circ
34. mLINO=60 m \angle LINO = 60^\circ

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