Question

these figures are congruent. what is m∠o?
□°

Explanation:

Step1: Recall Congruent Figures Property

Congruent figures have corresponding angles equal. So, we need to find the corresponding angle to ∠O in the first quadrilateral (FGHE).

Step2: Identify Corresponding Angle

In quadrilateral FGHE, the angles are 70° (∠F), 74° (∠H), 74° (∠E), and we can find ∠G by using the sum of interior angles of a quadrilateral (360°). Let's calculate ∠G: \(360 - 70 - 74 - 74 = 142\)°. Wait, no—wait, in the second figure (MNO P), ∠P is 142°, ∠M is 70°, ∠N is 74°, ∠O is what we need. Wait, actually, looking at the sides: HO corresponds to... Wait, the side lengths: 16 in (H to O? Wait, no, FGHE has side GH=16, HE, EF, FG. MNO P has side PO=16, MN=42, MP=42, NO=49. So corresponding angles: ∠O should correspond to ∠G? Wait no, wait let's check the angles. Wait, in FGHE, angles: ∠F=70°, ∠H=74°, ∠E=74°, so ∠G=360 - 70 -74 -74 = 142? No, wait no—wait the second figure: ∠M=70°, ∠N=74°, ∠P=142°, so ∠O should correspond to ∠G? Wait no, wait the problem is that the two figures are congruent, so their corresponding angles are equal. Wait, maybe I mixed up. Wait, let's list the angles of the second quadrilateral (MNO P): ∠M=70°, ∠N=74°, ∠P=142°, so ∠O is the remaining angle? Wait no, quadrilateral sum is 360. Wait 70 +74 +142 + ∠O = 360? 70+74=144, 144+142=286, 360-286=74? No, that can't be. Wait no, wait the first quadrilateral: FGHE, angles: ∠F=70°, ∠H=74°, ∠E=74°, so ∠G=360 -70 -74 -74 = 142°. Now, the second quadrilateral: MNO P, angles: ∠M=70° (corresponds to ∠F), ∠N=74° (corresponds to ∠H or ∠E), ∠P=142° (corresponds to ∠G), so ∠O should correspond to ∠E? Wait no, wait the side PO is 16, which corresponds to GH=16. So ∠O corresponds to ∠G? Wait no, maybe I made a mistake. Wait, no—wait the key is that congruent figures have corresponding angles equal. So looking at the first quadrilateral (FGHE), the angles are 70° (F), 74° (H), 74° (E), and the fourth angle (G) is 360 -70 -74 -74 = 142? No, 70+74=144, 144+74=218, 360-218=142. So ∠G=142°. Now, in the second quadrilateral (MNO P), ∠P=142°, ∠M=70°, ∠N=74°, so ∠O should be equal to ∠E, which is 74°? Wait no, that contradicts. Wait, no—wait maybe the correspondence is ∠O corresponds to ∠G? Wait no, let's check the side lengths. GH=16, PO=16. HE and NO: HE is a side, NO is 49. So GH (16) corresponds to PO (16). So angle at H (74°) corresponds to angle at O? No, wait ∠H is 74°, ∠O—wait no, maybe I messed up. Wait, the problem is that the two figures are congruent, so their corresponding angles are equal. Let's look at the first quadrilateral: angles are 70° (F), 74° (H), 74° (E), and ∠G. The second quadrilateral: angles are 70° (M), 74° (N), 142° (P), and ∠O. So if we match ∠F (70°) with ∠M (70°), ∠H (74°) with ∠N (74°), ∠P (142°) with ∠G, then ∠O should match ∠E (74°)? Wait no, that can't be. Wait, no—wait the sum of angles in a quadrilateral is 360. So for the first quadrilateral: 70 +74 +74 + x = 360 → x=142. So ∠G=142°. For the second quadrilateral: 70 (∠M) +74 (∠N) +142 (∠P) + ∠O = 360 → 70+74=144, 144+142=286, 360-286=74. Wait, that's 74. But that contradicts. Wait, no—wait maybe the correspondence is different. Wait, the side PO is 16, which is equal to GH=16. So ∠O is adjacent to PO=16, just like ∠G is adjacent to GH=16. Wait, in FGHE, GH is between ∠H and ∠G. In MNO P, PO is between ∠P and ∠O. So ∠G (adjacent to GH=16) should correspond to ∠O (adjacent to PO=16). Wait, ∠G is 142°, so ∠O should be 142°? Wait, no—wait I think I made a mistake earlier. Let's recalculate ∠G: 360 -70 (∠F) -74 (∠H) -74 (∠E) = 360 -218 = 142. So ∠G=142°. Now, in the second quadrilateral, ∠P=142°, ∠M=70°, ∠N=74°, so ∠O should be equal to ∠E? No, that's 74. Wait, this is confusing. Wait, the problem says the figures are congruent, so their corresponding angles are equal. So let's look at the angles: in the first figure, ∠F=70°, ∠H=74°, ∠E=74°, so the remaining angle (∠G) is 142°. In the second figure, ∠M=70°, ∠N=74°, ∠P=142°, so the remaining angle (∠O) should be equal to ∠E, which is 74°? Wait, no, that can't be. Wait, no—wait maybe the correspondence is ∠O corresponds to ∠G? Wait, no, the side lengths: GH=16, PO=16. HE and NO: HE is a side, NO is 49. So GH (16) and PO (16) are corresponding sides, so the angles adjacent to them: ∠H (74°) and ∠N (74°) are corresponding, ∠G and ∠O are corresponding. So ∠G=142°, so ∠O=142°? Wait, that makes sense. Wait, I think I messed up the sum earlier. Wait, 70 (∠F) +74 (∠H) +74 (∠E) +142 (∠G) = 70+74=144, 144+74=218, 218+142=360. Correct. For the second quadrilateral: 70 (∠M) +74 (∠N) +142 (∠P) + ∠O = 360? No, wait ∠P is 142, ∠M is 70, ∠N is 74, so ∠O should be 74? No, that's not adding up. Wait, no—wait the second figure's angles: ∠M=70°, ∠N=74°, ∠O=?, ∠P=142°. So 70+74+∠O+142=360 → ∠O=360 -70 -74 -142=74? But that's the same as ∠H and ∠E. But in the first figure, ∠G is 142, which should correspond to ∠P (142). So ∠O should correspond to ∠E (74°). Wait, maybe the labels are different. Let's look at the sides: GH=16, PO=16. HE and NO: HE is a side, NO is 49. FG and MN: FG and MN are 42? Wait, MN=42, MP=42. So FG corresponds to MN (42), MP corresponds to FG? Wait, maybe the correspondence is F→M, G→N, H→O, E→P? No, that doesn't make sense. Wait, the key is that congruent figures have corresponding angles equal, so we can find the angle in the first quadrilateral that corresponds to ∠O. Looking at the first quadrilateral (FGHE), the angle at E is 74°, angle at H is 74°, angle at F is 70°, so angle at G is 142°. In the second quadrilateral (MNO P), angle at M is 70°, angle at N is 74°, angle at P is 142°, so angle at O must be 74°? Wait, no, that can't be. Wait, I think I made a mistake in the correspondence. Wait, the problem is asking for m∠O. Let's look at the first figure: angles are ∠F=70°, ∠H=74°, ∠E=74°, so ∠G=360 -70 -74 -74=142°. Now, the second figure: ∠M=70°, ∠N=74°, ∠P=142°, so ∠O=360 -70 -74 -142=74°? But that's 74. But wait, maybe the corresponding angle to ∠O is ∠E, which is 74°. So m∠O=74? Wait, no, that seems conflicting. Wait, no—wait let's check again. The sum of angles in a quadrilateral is 360. So for the first quadrilateral: 70 +74 +74 + x = 360 → x=142. So ∠G=142. For the second quadrilateral: 70 (∠M) +74 (∠N) +142 (∠P) + ∠O = 360 → 70+74=144, 144+142=286, 360-286=74. So ∠O=74. But that's the same as ∠H and ∠E. So maybe that's correct. Wait, but why is ∠P=142? Because ∠P corresponds to ∠G (142°). So ∠O corresponds to ∠E (74°). So the answer is 74? Wait, no—wait I think I messed up. Wait, the first figure: vertices F, G, H, E. The second figure: M, N, O, P. So the order is F-G-H-E and M-N-O-P. So corresponding angles: F→M, G→N, H→O, E→P. Oh! That's the key. So ∠H corresponds to ∠O. ∠H is 74°, so ∠O is 74°. Yes! That makes sense. So the correspondence is F to M, G to N, H to O, E to P. So ∠H (74°) corresponds to ∠O, so m∠O=74°.

Answer:

74