WEBVTT 00:00:02.140 --> 00:00:08.790 Is the line segment between the midpoints of two sides of a triangle parallel to the other side? 00:00:09.640 --> 00:00:15.720 To answer this question, let’s take a triangle, which we can call triangle 𝐴𝐡𝐢. 00:00:16.550 --> 00:00:25.330 We want to consider the line segment between the midpoints of two sides of the triangle, which is also called a midsegment. 00:00:25.890 --> 00:00:32.850 We can draw 𝐷 and 𝐸 at the midpoints of line segments 𝐴𝐡 and 𝐴𝐢. 00:00:33.540 --> 00:00:47.170 And as these are midpoints, we know that there will be two pairs of congruent line segments, since 𝐴𝐷 and 𝐷𝐡 are congruent and 𝐴𝐸 and 𝐸𝐢 are congruent. 00:00:47.810 --> 00:00:55.720 We can join the midpoints 𝐷 and 𝐸 to create the midsegment, which will be line segment 𝐷𝐸. 00:00:56.590 --> 00:01:06.090 We now need to determine if the midsegment 𝐷𝐸 is parallel to the third side of the triangle, which is line segment 𝐡𝐢. 00:01:06.660 --> 00:01:13.270 The important first step we take in working out the answer is to draw a new line. 00:01:13.780 --> 00:01:25.900 We construct the line passing through 𝐢 and a new point 𝑃 such that this new line 𝐢𝑃 is parallel to the line segment 𝐴𝐡. 00:01:26.620 --> 00:01:35.320 We can then extend the line segment 𝐷𝐸 so that it meets the new line 𝐢𝑃 at the point 𝐹. 00:01:36.180 --> 00:01:44.280 Now let’s consider some of the sides and angles in the triangles 𝐴𝐸𝐷 and 𝐢𝐸𝐹. 00:01:45.120 --> 00:01:48.920 Firstly, we know that we have two congruent sides. 00:01:49.490 --> 00:01:57.480 Since we set 𝐸 to be the midpoint of line segment 𝐴𝐢, we can write that 𝐴𝐸 equals 𝐢𝐸. 00:01:58.330 --> 00:02:09.130 Then, because we have a pair of parallel lines and a transversal line segment 𝐴𝐢, we can identify a pair of alternate interior angles. 00:02:09.920 --> 00:02:18.640 As these angles are congruent, we can write that the measures of angles 𝐷𝐴𝐸 and 𝐹𝐢𝐸 are equal. 00:02:19.260 --> 00:02:26.740 We can also observe that angles 𝐷𝐸𝐴 and 𝐹𝐸𝐢 are vertically opposite angles. 00:02:27.280 --> 00:02:30.190 So their measures are also equal. 00:02:30.900 --> 00:02:46.650 And so because we have two pairs of congruent angles and an included pair of sides congruent, we can write that triangles 𝐴𝐸𝐷 and 𝐢𝐸𝐹 are congruent by the ASA criterion. 00:02:47.510 --> 00:02:57.510 We can notice at this point that even with these congruent triangles, we still haven’t determined if we have a pair of parallel lines. 00:02:58.100 --> 00:03:03.230 But let’s think about what we can determine from these congruent triangles. 00:03:03.810 --> 00:03:11.100 We can identify that the corresponding sides 𝐴𝐷 and 𝐢𝐹 must be congruent. 00:03:11.580 --> 00:03:18.270 And of course we already created the line segment 𝐴𝐡 such that 𝐷 is the midpoint. 00:03:18.550 --> 00:03:22.600 So 𝐡𝐷 is also congruent to 𝐴𝐷. 00:03:23.090 --> 00:03:27.870 Now we know that in the figure, there are three congruent line segments. 00:03:28.810 --> 00:03:33.720 The final step is to consider the polygon 𝐡𝐢𝐹𝐷. 00:03:34.450 --> 00:03:41.830 𝐡𝐢𝐹𝐷 is a quadrilateral, and we have shown that the sides 𝐡𝐷 and 𝐹𝐢 are congruent. 00:03:42.260 --> 00:03:49.250 And we know that they are also parallel because we constructed the line 𝐢𝑃 to be parallel. 00:03:49.780 --> 00:03:57.020 This property is sufficient to prove that the shape 𝐡𝐢𝐹𝐷 is a parallelogram. 00:03:57.490 --> 00:04:03.920 And we know that parallelograms have two pairs of opposite sides parallel. 00:04:04.650 --> 00:04:10.200 So line segments 𝐷𝐹 and 𝐡𝐢 are also parallel. 00:04:11.020 --> 00:04:20.110 We can therefore return to the original question to see if the midsegment of a triangle is parallel to the third side. 00:04:20.570 --> 00:04:28.480 And the answer is yes, because the line segment 𝐷𝐸 is contained within the line segment 𝐷𝐹. 00:04:29.020 --> 00:04:36.680 So either line segment 𝐷𝐸 or 𝐷𝐹 can be said to be parallel to line segment 𝐡𝐢. 00:04:37.250 --> 00:04:44.460 In fact, this property that we have just proved is one of the triangle midsegment theorems. 00:04:45.110 --> 00:04:54.000 It can be stated as the line segment joining the midpoints of two sides of a triangle is parallel to the third side. 00:04:54.490 --> 00:04:59.970 We can learn this property and apply it directly in many different problems. 00:05:00.810 --> 00:05:07.510 Stating this theorem would be sufficient for us to give the answer to the question as yes.